hdmapval2lem - Mathbox for Norm Megill

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Theorem hdmapval2lem 39845

Description: Lemma for hdmapval2 39846. (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 39842	. . 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 2740	. 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 2738	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
16		eqid 2738	. . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
17		eqid 2738	. . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
18		eqid 2738	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		eqid 2738	. . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
20	1, 18, 19, 3, 4, 15, 2, 11	dvheveccl 39126	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
21	1, 3, 4, 15, 5, 6, 16, 17, 8, 11, 20	mapdhvmap 39783	. . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)}))
22		eqid 2738	. . . . . 6 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
23	1, 3, 4, 15, 6, 7, 22, 8, 11, 20	hvmapcl2 39780	. . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0_g‘𝐶)}))
24	23	eldifad 3899	. . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷)
25	1, 3, 4, 15, 5, 6, 7, 16, 17, 9, 11, 21, 20, 24, 12	hdmap1eu 39838	. . 3 ⊢ (𝜑 → ∃!ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)))
26		nfv 1917	. . . 4 ⊢ Ⅎℎ𝜑
27		nfcvd 2908	. . . 4 ⊢ (𝜑 → Ⅎℎ𝐹)
28		nfvd 1918	. . . 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 2742	. . . . . . 7 ⊢ (ℎ = 𝐹 → (ℎ = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩) ↔ 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)))
31	30	imbi2d 341	. . . . . 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 3112	. . . . 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 482	. . . 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 7256	. . 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 684	. 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 281	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃!wreu 3066 ∪ cun 3885 {csn 4561 ⟨cop 4567 ⟨cotp 4569 I cid 5488 ↾ cres 5591 ‘cfv 6433 ℩crio 7231 Basecbs 16912 0_gc0g 17150 LSpanclspn 20233 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 DVecHcdvh 39092 LCDualclcd 39600 mapdcmpd 39638 HVMapchvm 39770 HDMap1chdma1 39805 HDMapchdma 39806

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lcv 37033 df-lfl 37072 df-lkr 37100 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409 df-lcdual 39601 df-mapd 39639 df-hvmap 39771 df-hdmap1 39807 df-hdmap 39808

This theorem is referenced by: hdmapval2 39846

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