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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapval2lem | Structured version Visualization version GIF version |
Description: Lemma for hdmapval2 37986. (Contributed by NM, 15-May-2015.) |
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) |
Ref | Expression |
---|---|
hdmapval2lem | ⊢ (𝜑 → ((𝑆‘𝑇) = 𝐹 ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) |
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 37982 | . . 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 2780 | . 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 2778 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
16 | eqid 2778 | . . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
17 | eqid 2778 | . . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
18 | eqid 2778 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
19 | eqid 2778 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
20 | 1, 18, 19, 3, 4, 15, 2, 11 | dvheveccl 37266 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
21 | 1, 3, 4, 15, 5, 6, 16, 17, 8, 11, 20 | mapdhvmap 37923 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)})) |
22 | eqid 2778 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
23 | 1, 3, 4, 15, 6, 7, 22, 8, 11, 20 | hvmapcl2 37920 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
24 | 23 | eldifad 3804 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
25 | 1, 3, 4, 15, 5, 6, 7, 16, 17, 9, 11, 21, 20, 24, 12 | hdmap1eu 37978 | . . 3 ⊢ (𝜑 → ∃!ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉))) |
26 | nfv 1957 | . . . 4 ⊢ Ⅎℎ𝜑 | |
27 | nfcvd 2935 | . . . 4 ⊢ (𝜑 → Ⅎℎ𝐹) | |
28 | nfvd 1958 | . . . 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 2782 | . . . . . . 7 ⊢ (ℎ = 𝐹 → (ℎ = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) ↔ 𝐹 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉))) | |
31 | 30 | imbi2d 332 | . . . . . 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 3168 | . . . . 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 475 | . . . 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 6903 | . . 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 677 | . 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 274 | 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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∃!wreu 3092 ∪ cun 3790 {csn 4398 〈cop 4404 〈cotp 4406 I cid 5260 ↾ cres 5357 ‘cfv 6135 ℩crio 6882 Basecbs 16255 0gc0g 16486 LSpanclspn 19366 HLchlt 35504 LHypclh 36138 LTrncltrn 36255 DVecHcdvh 37232 LCDualclcd 37740 mapdcmpd 37778 HVMapchvm 37910 HDMap1chdma1 37945 HDMapchdma 37946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35107 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35130 df-lshyp 35131 df-lcv 35173 df-lfl 35212 df-lkr 35240 df-ldual 35278 df-oposet 35330 df-ol 35332 df-oml 35333 df-covers 35420 df-ats 35421 df-atl 35452 df-cvlat 35476 df-hlat 35505 df-llines 35652 df-lplanes 35653 df-lvols 35654 df-lines 35655 df-psubsp 35657 df-pmap 35658 df-padd 35950 df-lhyp 36142 df-laut 36143 df-ldil 36258 df-ltrn 36259 df-trl 36313 df-tgrp 36897 df-tendo 36909 df-edring 36911 df-dveca 37157 df-disoa 37183 df-dvech 37233 df-dib 37293 df-dic 37327 df-dih 37383 df-doch 37502 df-djh 37549 df-lcdual 37741 df-mapd 37779 df-hvmap 37911 df-hdmap1 37947 df-hdmap 37948 |
This theorem is referenced by: hdmapval2 37986 |
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