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Theorem cnlnadjlem9 31903

Description: Lemma for cnlnadji 31904. 𝐹 provides an example showing the existence of a continuous linear adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cnlnadjlem.1	⊢ 𝑇 ∈ LinOp
cnlnadjlem.2	⊢ 𝑇 ∈ ContOp
cnlnadjlem.3	⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·_ih 𝑦))
cnlnadjlem.4	⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤))
cnlnadjlem.5	⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)

**Assertion**
Ref	Expression
cnlnadjlem9	⊢ ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))

Distinct variable groups: 𝑣,𝑔,𝑤,𝑦,𝑧 𝑧,𝐵 𝑤,𝑡,𝑥,𝑧,𝐹 𝑡,𝑔,𝑥,𝑇,𝑣,𝑦,𝑤,𝑧 𝑣,𝐺,𝑤,𝑥,𝑧 Allowed substitution hints: 𝐵(𝑥,𝑦,𝑤,𝑣,𝑡,𝑔) 𝐹(𝑦,𝑣,𝑔) 𝐺(𝑦,𝑡,𝑔)

**Proof of Theorem cnlnadjlem9**
Step	Hyp	Ref	Expression
1		cnlnadjlem.1	. . . 4 ⊢ 𝑇 ∈ LinOp
2		cnlnadjlem.2	. . . 4 ⊢ 𝑇 ∈ ContOp
3		cnlnadjlem.3	. . . 4 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·_ih 𝑦))
4		cnlnadjlem.4	. . . 4 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤))
5		cnlnadjlem.5	. . . 4 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)
6	1, 2, 3, 4, 5	cnlnadjlem6 31900	. . 3 ⊢ 𝐹 ∈ LinOp
7	1, 2, 3, 4, 5	cnlnadjlem8 31902	. . 3 ⊢ 𝐹 ∈ ContOp
8		elin 3963	. . 3 ⊢ (𝐹 ∈ (LinOp ∩ ContOp) ↔ (𝐹 ∈ LinOp ∧ 𝐹 ∈ ContOp))
9	6, 7, 8	mpbir2an 709	. 2 ⊢ 𝐹 ∈ (LinOp ∩ ContOp)
10	1, 2, 3, 4, 5	cnlnadjlem5 31899	. . . 4 ⊢ ((𝑧 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝐹‘𝑧)))
11	10	ancoms 457	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝐹‘𝑧)))
12	11	rgen2 3193	. 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝐹‘𝑧))
13		fveq1 6899	. . . . . 6 ⊢ (𝑡 = 𝐹 → (𝑡‘𝑧) = (𝐹‘𝑧))
14	13	oveq2d 7440	. . . . 5 ⊢ (𝑡 = 𝐹 → (𝑥 ·_ih (𝑡‘𝑧)) = (𝑥 ·_ih (𝐹‘𝑧)))
15	14	eqeq2d 2738	. . . 4 ⊢ (𝑡 = 𝐹 → (((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) ↔ ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝐹‘𝑧))))
16	15	2ralbidv 3214	. . 3 ⊢ (𝑡 = 𝐹 → (∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝐹‘𝑧))))
17	16	rspcev 3609	. 2 ⊢ ((𝐹 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝐹‘𝑧))) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)))
18	9, 12, 17	mp2an 690	1 ⊢ ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 ∀wral 3057 ∃wrex 3066 ∩ cin 3946 ↦ cmpt 5233 ‘cfv 6551 ℩crio 7379 (class class class)co 7424 ℋchba 30747 ·_ih csp 30750 ContOpccop 30774 LinOpclo 30775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cc 10464 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223 ax-mulf 11224 ax-hilex 30827 ax-hfvadd 30828 ax-hvcom 30829 ax-hvass 30830 ax-hv0cl 30831 ax-hvaddid 30832 ax-hfvmul 30833 ax-hvmulid 30834 ax-hvmulass 30835 ax-hvdistr1 30836 ax-hvdistr2 30837 ax-hvmul0 30838 ax-hfi 30907 ax-his1 30910 ax-his2 30911 ax-his3 30912 ax-his4 30913 ax-hcompl 31030

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-oadd 8495 df-omul 8496 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-acn 9971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ioo 13366 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-fl 13795 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-rlim 15471 df-sum 15671 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-cn 23149 df-cnp 23150 df-lm 23151 df-t1 23236 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cfil 25201 df-cau 25202 df-cmet 25203 df-grpo 30321 df-gid 30322 df-ginv 30323 df-gdiv 30324 df-ablo 30373 df-vc 30387 df-nv 30420 df-va 30423 df-ba 30424 df-sm 30425 df-0v 30426 df-vs 30427 df-nmcv 30428 df-ims 30429 df-dip 30529 df-ssp 30550 df-ph 30641 df-cbn 30691 df-hnorm 30796 df-hba 30797 df-hvsub 30799 df-hlim 30800 df-hcau 30801 df-sh 31035 df-ch 31049 df-oc 31080 df-ch0 31081 df-nmop 31667 df-cnop 31668 df-lnop 31669 df-unop 31671 df-nmfn 31673 df-nlfn 31674 df-cnfn 31675 df-lnfn 31676

This theorem is referenced by: cnlnadji 31904

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