hdmapval2lem - Mathbox for Norm Megill

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Theorem hdmapval2lem 41950

Description: Lemma for hdmapval2 41951. (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 41947	. . 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 2735	. 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 2733	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
16		eqid 2733	. . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
17		eqid 2733	. . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
18		eqid 2733	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		eqid 2733	. . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
20	1, 18, 19, 3, 4, 15, 2, 11	dvheveccl 41231	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
21	1, 3, 4, 15, 5, 6, 16, 17, 8, 11, 20	mapdhvmap 41888	. . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)}))
22		eqid 2733	. . . . . 6 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
23	1, 3, 4, 15, 6, 7, 22, 8, 11, 20	hvmapcl2 41885	. . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0_g‘𝐶)}))
24	23	eldifad 3910	. . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷)
25	1, 3, 4, 15, 5, 6, 7, 16, 17, 9, 11, 21, 20, 24, 12	hdmap1eu 41943	. . 3 ⊢ (𝜑 → ∃!ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)))
26		nfv 1915	. . . 4 ⊢ Ⅎℎ𝜑
27		nfcvd 2896	. . . 4 ⊢ (𝜑 → Ⅎℎ𝐹)
28		nfvd 1916	. . . 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 2737	. . . . . . 7 ⊢ (ℎ = 𝐹 → (ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩) ↔ 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)))
31	30	imbi2d 340	. . . . . 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 3156	. . . . 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 481	. . . 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 7332	. . 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 687	. 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 282	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3048 ∃!wreu 3345 ∪ cun 3896 {csn 4575 ⟨cop 4581 ⟨cotp 4583 I cid 5513 ↾ cres 5621 ‘cfv 6486 ℩crio 7308 Basecbs 17122 0_gc0g 17345 LSpanclspn 20906 HLchlt 39469 LHypclh 40103 LTrncltrn 40220 DVecHcdvh 41197 LCDualclcd 41705 mapdcmpd 41743 HVMapchvm 41875 HDMap1chdma1 41910 HDMapchdma 41911

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-riotaBAD 39072

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-ot 4584 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-tpos 8162 df-undef 8209 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-n0 12389 df-z 12476 df-uz 12739 df-fz 13410 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-sca 17179 df-vsca 17180 df-0g 17347 df-mre 17490 df-mrc 17491 df-acs 17493 df-proset 18202 df-poset 18221 df-plt 18236 df-lub 18252 df-glb 18253 df-join 18254 df-meet 18255 df-p0 18331 df-p1 18332 df-lat 18340 df-clat 18407 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-subg 19038 df-cntz 19231 df-oppg 19260 df-lsm 19550 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-nzr 20430 df-rlreg 20611 df-domn 20612 df-drng 20648 df-lmod 20797 df-lss 20867 df-lsp 20907 df-lvec 21039 df-lsatoms 39095 df-lshyp 39096 df-lcv 39138 df-lfl 39177 df-lkr 39205 df-ldual 39243 df-oposet 39295 df-ol 39297 df-oml 39298 df-covers 39385 df-ats 39386 df-atl 39417 df-cvlat 39441 df-hlat 39470 df-llines 39617 df-lplanes 39618 df-lvols 39619 df-lines 39620 df-psubsp 39622 df-pmap 39623 df-padd 39915 df-lhyp 40107 df-laut 40108 df-ldil 40223 df-ltrn 40224 df-trl 40278 df-tgrp 40862 df-tendo 40874 df-edring 40876 df-dveca 41122 df-disoa 41148 df-dvech 41198 df-dib 41258 df-dic 41292 df-dih 41348 df-doch 41467 df-djh 41514 df-lcdual 41706 df-mapd 41744 df-hvmap 41876 df-hdmap1 41912 df-hdmap 41913

This theorem is referenced by: hdmapval2 41951

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