dih1dimb2 - Mathbox for Norm Megill

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Theorem dih1dimb2 40714

Description: Isomorphism H at an atom under 𝑊. (Contributed by NM, 27-Apr-2014.)

**Hypotheses**
Ref	Expression
dih1dimb2.b	⊢ 𝐵 = (Base‘𝐾)
dih1dimb2.l	⊢ ≤ = (le‘𝐾)
dih1dimb2.a	⊢ 𝐴 = (Atoms‘𝐾)
dih1dimb2.h	⊢ 𝐻 = (LHyp‘𝐾)
dih1dimb2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dih1dimb2.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dih1dimb2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dih1dimb2.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dih1dimb2.n	⊢ 𝑁 = (LSpan‘𝑈)

**Assertion**
Ref	Expression
dih1dimb2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))

Distinct variable groups: ≤ ,𝑓 𝐴,𝑓 𝐵,ℎ 𝑓,𝐻 𝑓,ℎ,𝐾 𝑄,𝑓 𝑇,𝑓,ℎ 𝑓,𝑊,ℎ Allowed substitution hints: 𝐴(ℎ) 𝐵(𝑓) 𝑄(ℎ) 𝑈(𝑓,ℎ) 𝐻(ℎ) 𝐼(𝑓,ℎ) ≤ (ℎ) 𝑁(𝑓,ℎ) 𝑂(𝑓,ℎ)

**Proof of Theorem dih1dimb2**
Step	Hyp	Ref	Expression
1		dih1dimb2.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		dih1dimb2.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		dih1dimb2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		dih1dimb2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		eqid 2728	. . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	cdlemf 40036	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄)
7		simp3 1136	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄)
8		simp1rl 1236	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑄 ∈ 𝐴)
9	7, 8	eqeltrd 2829	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴)
10		simp1l 1195	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		simp2 1135	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ∈ 𝑇)
12		dih1dimb2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
13	12, 2, 3, 4, 5	trlnidatb 39650	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴))
14	10, 11, 13	syl2anc 583	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴))
15	9, 14	mpbird 257	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ≠ ( I ↾ 𝐵))
16	7	fveq2d 6901	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝐼‘𝑄))
17		dih1dimb2.o	. . . . . . . 8 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
18		dih1dimb2.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
19		dih1dimb2.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
20		dih1dimb2.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
21	12, 3, 4, 5, 17, 18, 19, 20	dih1dimb 40713	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{⟨𝑓, 𝑂⟩}))
22	10, 11, 21	syl2anc 583	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{⟨𝑓, 𝑂⟩}))
23	16, 22	eqtr3d 2770	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))
24	15, 23	jca 511	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))
25	24	3expia 1119	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄 → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))))
26	25	reximdva 3165	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄 → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))))
27	6, 26	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∃wrex 3067 {csn 4629 ⟨cop 4635 class class class wbr 5148 ↦ cmpt 5231 I cid 5575 ↾ cres 5680 ‘cfv 6548 Basecbs 17179 lecple 17239 LSpanclspn 20854 Atomscatm 38735 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 trLctrl 39631 DVecHcdvh 40551 DIsoHcdih 40701

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38425

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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-undef 8278 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-drng 20625 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lvec 20987 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tendo 40228 df-edring 40230 df-disoa 40502 df-dvech 40552 df-dib 40612 df-dih 40702

This theorem is referenced by: dihatlat 40807 dihatexv 40811

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