mdbr2 - Hilbert Space Explorer

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Theorem mdbr2 32099

Description: Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 32097. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

**Assertion**
Ref	Expression
mdbr2	⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝐴 𝑀_ℋ 𝐵 ↔ ∀𝑥 ∈ C_ℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem mdbr2**
Step	Hyp	Ref	Expression
1		mdbr 32097	. 2 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝐴 𝑀_ℋ 𝐵 ↔ ∀𝑥 ∈ C_ℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
2		chub1 31310	. . . . . . . . . 10 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → 𝑥 ⊆ (𝑥 ∨_ℋ 𝐴))
3	2	ancoms 458	. . . . . . . . 9 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → 𝑥 ⊆ (𝑥 ∨_ℋ 𝐴))
4		iba 527	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ↔ (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵)))
5		ssin 4226	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
6	4, 5	bitrdi 287	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ↔ 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
7	3, 6	syl5ibcom 244	. . . . . . . 8 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝑥 ⊆ 𝐵 → 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
8		chub2 31311	. . . . . . . . 9 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → 𝐴 ⊆ (𝑥 ∨_ℋ 𝐴))
9	8	ssrind 4231	. . . . . . . 8 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
10	7, 9	jctird 526	. . . . . . 7 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝑥 ⊆ 𝐵 → (𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))))
11	10	adantlr 714	. . . . . 6 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) ∧ 𝑥 ∈ C_ℋ ) → (𝑥 ⊆ 𝐵 → (𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))))
12		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) ∧ 𝑥 ∈ C_ℋ ) → 𝑥 ∈ C_ℋ )
13		chincl 31302	. . . . . . . 8 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝐴 ∩ 𝐵) ∈ C_ℋ )
14	13	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) ∧ 𝑥 ∈ C_ℋ ) → (𝐴 ∩ 𝐵) ∈ C_ℋ )
15		chjcl 31160	. . . . . . . . . 10 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐴) ∈ C_ℋ )
16	15	ancoms 458	. . . . . . . . 9 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐴) ∈ C_ℋ )
17		chincl 31302	. . . . . . . . 9 ⊢ (((𝑥 ∨_ℋ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ )
18	16, 17	sylan 579	. . . . . . . 8 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) ∧ 𝐵 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ )
19	18	an32s 651	. . . . . . 7 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) ∧ 𝑥 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ )
20		chlub 31312	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐴 ∩ 𝐵) ∈ C_ℋ ∧ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ ) → ((𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)) ↔ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
21	12, 14, 19, 20	syl3anc 1369	. . . . . 6 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) ∧ 𝑥 ∈ C_ℋ ) → ((𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)) ↔ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
22	11, 21	sylibd 238	. . . . 5 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) ∧ 𝑥 ∈ C_ℋ ) → (𝑥 ⊆ 𝐵 → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
23		eqss 3993	. . . . . 6 ⊢ (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ∧ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
24	23	rbaib 538	. . . . 5 ⊢ ((𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) → (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))))
25	22, 24	syl6 35	. . . 4 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) ∧ 𝑥 ∈ C_ℋ ) → (𝑥 ⊆ 𝐵 → (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
26	25	pm5.74d 273	. . 3 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) ∧ 𝑥 ∈ C_ℋ ) → ((𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))) ↔ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
27	26	ralbidva 3171	. 2 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (∀𝑥 ∈ C_ℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))) ↔ ∀𝑥 ∈ C_ℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
28	1, 27	bitrd 279	1 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝐴 𝑀_ℋ 𝐵 ↔ ∀𝑥 ∈ C_ℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∩ cin 3944 ⊆ wss 3945 class class class wbr 5142 (class class class)co 7414 C_ℋ cch 30732 ∨_ℋ chj 30736 𝑀_ℋ cmd 30769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cc 10452 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 ax-mulf 11212 ax-hilex 30802 ax-hfvadd 30803 ax-hvcom 30804 ax-hvass 30805 ax-hv0cl 30806 ax-hvaddid 30807 ax-hfvmul 30808 ax-hvmulid 30809 ax-hvmulass 30810 ax-hvdistr1 30811 ax-hvdistr2 30812 ax-hvmul0 30813 ax-hfi 30882 ax-his1 30885 ax-his2 30886 ax-his3 30887 ax-his4 30888 ax-hcompl 31005

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-acn 9959 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-rlim 15459 df-sum 15659 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-cn 23124 df-cnp 23125 df-lm 23126 df-haus 23212 df-tx 23459 df-hmeo 23652 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-xms 24219 df-ms 24220 df-tms 24221 df-cfil 25176 df-cau 25177 df-cmet 25178 df-grpo 30296 df-gid 30297 df-ginv 30298 df-gdiv 30299 df-ablo 30348 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-vs 30402 df-nmcv 30403 df-ims 30404 df-dip 30504 df-ssp 30525 df-ph 30616 df-cbn 30666 df-hnorm 30771 df-hba 30772 df-hvsub 30774 df-hlim 30775 df-hcau 30776 df-sh 31010 df-ch 31024 df-oc 31055 df-ch0 31056 df-shs 31111 df-chj 31113 df-md 32083

This theorem is referenced by: mdbr4 32101 mdsl0 32113 ssmd1 32114 ssmd2 32115 mdslmd1i 32132 mdslmd3i 32135 mdexchi 32138

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