hdmapval2lem - Mathbox for Norm Megill

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Theorem hdmapval2lem 42646

Description: Lemma for hdmapval2 42647. (Contributed by NM, 15-May-2015.)

**Hypotheses**
Ref	Expression
hdmapval2.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapval2.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapval2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapval2.v	⊢ 𝑉 = (Base‘𝑈)
hdmapval2.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmapval2.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmapval2.d	⊢ 𝐷 = (Base‘𝐶)
hdmapval2.j	⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
hdmapval2.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmapval2.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapval2.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapval2.t	⊢ (𝜑 → 𝑇 ∈ 𝑉)
hdmapval2.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)

**Assertion**
Ref	Expression
hdmapval2lem	⊢ (𝜑 → ((𝑆‘𝑇) = 𝐹 ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))

Distinct variable groups: 𝑧,𝐶 𝑧,𝐷 𝑧,𝐸 𝑧,𝐹 𝑧,𝐼 𝑧,𝐽 𝑧,𝐾 𝑧,𝑁 𝑧,𝑇 𝑧,𝑈 𝑧,𝑉 𝑧,𝑊 𝜑,𝑧 Allowed substitution hints: 𝑆(𝑧) 𝐻(𝑧)

Proof of Theorem hdmapval2lem Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapval2.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmapval2.e	. . . 4 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
3		hdmapval2.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		hdmapval2.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		hdmapval2.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
6		hdmapval2.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		hdmapval2.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
8		hdmapval2.j	. . . 4 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
9		hdmapval2.i	. . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
10		hdmapval2.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
11		hdmapval2.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		hdmapval2.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	hdmapval 42643	. . 3 ⊢ (𝜑 → (𝑆‘𝑇) = (℩ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
14	13	eqeq1d 2768	. 2 ⊢ (𝜑 → ((𝑆‘𝑇) = 𝐹 ↔ (℩ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))) = 𝐹))
15		eqid 2766	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
16		eqid 2766	. . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
17		eqid 2766	. . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
18		eqid 2766	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		eqid 2766	. . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
20	1, 18, 19, 3, 4, 15, 2, 11	dvheveccl 41927	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
21	1, 3, 4, 15, 5, 6, 16, 17, 8, 11, 20	mapdhvmap 42584	. . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)}))
22		eqid 2766	. . . . . 6 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
23	1, 3, 4, 15, 6, 7, 22, 8, 11, 20	hvmapcl2 42581	. . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0_g‘𝐶)}))
24	23	eldifad 3920	. . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷)
25	1, 3, 4, 15, 5, 6, 7, 16, 17, 9, 11, 21, 20, 24, 12	hdmap1eu 42639	. . 3 ⊢ (𝜑 → ∃!ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)))
26		nfv 1947	. . . 4 ⊢ Ⅎℎ𝜑
27		nfcvd 2929	. . . 4 ⊢ (𝜑 → Ⅎℎ𝐹)
28		nfvd 1948	. . . 4 ⊢ (𝜑 → Ⅎℎ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)))
29		hdmapval2.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
30		eqeq1 2770	. . . . . . 7 ⊢ (ℎ = 𝐹 → (ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩) ↔ 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)))
31	30	imbi2d 343	. . . . . 6 ⊢ (ℎ = 𝐹 → ((¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)) ↔ (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
32	31	ralbidv 3191	. . . . 5 ⊢ (ℎ = 𝐹 → (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)) ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
33	32	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐹) → (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)) ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
34	26, 27, 28, 29, 33	riota2df 7403	. . 3 ⊢ ((𝜑 ∧ ∃!ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))) → (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)) ↔ (℩ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))) = 𝐹))
35	25, 34	mpdan 700	. 2 ⊢ (𝜑 → (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)) ↔ (℩ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))) = 𝐹))
36	14, 35	bitr4d 285	1 ⊢ (𝜑 → ((𝑆‘𝑇) = 𝐹 ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3082 ∃!wreu 3370 ∪ cun 3906 {csn 4594 ⟨cop 4600 ⟨cotp 4602 I cid 5560 ↾ cres 5668 ‘cfv 6543 ℩crio 7379 Basecbs 17294 0_gc0g 17517 LSpanclspn 21129 HLchlt 40165 LHypclh 40799 LTrncltrn 40916 DVecHcdvh 41893 LCDualclcd 42401 mapdcmpd 42439 HVMapchvm 42571 HDMap1chdma1 42606 HDMapchdma 42607

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-riotaBAD 39768

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-ot 4603 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-tpos 8231 df-undef 8278 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-n0 12523 df-z 12610 df-uz 12881 df-fz 13554 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-sca 17351 df-vsca 17352 df-0g 17519 df-mre 17663 df-mrc 17664 df-acs 17666 df-proset 18375 df-poset 18394 df-plt 18409 df-lub 18425 df-glb 18426 df-join 18427 df-meet 18428 df-p0 18504 df-p1 18505 df-lat 18513 df-clat 18580 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-grp 19034 df-minusg 19035 df-sbg 19036 df-subg 19220 df-cntz 19418 df-oppg 19447 df-lsm 19737 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-invr 20503 df-dvr 20516 df-nzr 20647 df-rlreg 20830 df-domn 20831 df-drng 20866 df-lmod 21020 df-lss 21090 df-lsp 21130 df-lvec 21261 df-lsatoms 39791 df-lshyp 39792 df-lcv 39834 df-lfl 39873 df-lkr 39901 df-ldual 39939 df-oposet 39991 df-ol 39993 df-oml 39994 df-covers 40081 df-ats 40082 df-atl 40113 df-cvlat 40137 df-hlat 40166 df-llines 40313 df-lplanes 40314 df-lvols 40315 df-lines 40316 df-psubsp 40318 df-pmap 40319 df-padd 40611 df-lhyp 40803 df-laut 40804 df-ldil 40919 df-ltrn 40920 df-trl 40974 df-tgrp 41558 df-tendo 41570 df-edring 41572 df-dveca 41818 df-disoa 41844 df-dvech 41894 df-dib 41954 df-dic 41988 df-dih 42044 df-doch 42163 df-djh 42210 df-lcdual 42402 df-mapd 42440 df-hvmap 42572 df-hdmap1 42608 df-hdmap 42609

This theorem is used by: hdmapval2 42647

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