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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapval2lem | Structured version Visualization version GIF version |
Description: Lemma for hdmapval2 41309. (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 41305 | . . 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 2729 | . 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 2727 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
16 | eqid 2727 | . . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
17 | eqid 2727 | . . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
18 | eqid 2727 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
19 | eqid 2727 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
20 | 1, 18, 19, 3, 4, 15, 2, 11 | dvheveccl 40589 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
21 | 1, 3, 4, 15, 5, 6, 16, 17, 8, 11, 20 | mapdhvmap 41246 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)})) |
22 | eqid 2727 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
23 | 1, 3, 4, 15, 6, 7, 22, 8, 11, 20 | hvmapcl2 41243 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
24 | 23 | eldifad 3959 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
25 | 1, 3, 4, 15, 5, 6, 7, 16, 17, 9, 11, 21, 20, 24, 12 | hdmap1eu 41301 | . . 3 ⊢ (𝜑 → ∃!ℎ ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → ℎ = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉))) |
26 | nfv 1909 | . . . 4 ⊢ Ⅎℎ𝜑 | |
27 | nfcvd 2899 | . . . 4 ⊢ (𝜑 → Ⅎℎ𝐹) | |
28 | nfvd 1910 | . . . 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 2731 | . . . . . . 7 ⊢ (ℎ = 𝐹 → (ℎ = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉) ↔ 𝐹 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉))) | |
31 | 30 | imbi2d 339 | . . . . . 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 3173 | . . . . 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 480 | . . . 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 7404 | . . 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 685 | . 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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3057 ∃!wreu 3370 ∪ cun 3945 {csn 4630 〈cop 4636 〈cotp 4638 I cid 5577 ↾ cres 5682 ‘cfv 6551 ℩crio 7379 Basecbs 17185 0gc0g 17426 LSpanclspn 20860 HLchlt 38826 LHypclh 39461 LTrncltrn 39578 DVecHcdvh 40555 LCDualclcd 41063 mapdcmpd 41101 HVMapchvm 41233 HDMap1chdma1 41268 HDMapchdma 41269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-riotaBAD 38429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 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 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-tpos 8236 df-undef 8283 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-0g 17428 df-mre 17571 df-mrc 17572 df-acs 17574 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-p1 18423 df-lat 18429 df-clat 18496 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-subg 19083 df-cntz 19273 df-oppg 19302 df-lsm 19596 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-dvr 20345 df-drng 20631 df-lmod 20750 df-lss 20821 df-lsp 20861 df-lvec 20993 df-lsatoms 38452 df-lshyp 38453 df-lcv 38495 df-lfl 38534 df-lkr 38562 df-ldual 38600 df-oposet 38652 df-ol 38654 df-oml 38655 df-covers 38742 df-ats 38743 df-atl 38774 df-cvlat 38798 df-hlat 38827 df-llines 38975 df-lplanes 38976 df-lvols 38977 df-lines 38978 df-psubsp 38980 df-pmap 38981 df-padd 39273 df-lhyp 39465 df-laut 39466 df-ldil 39581 df-ltrn 39582 df-trl 39636 df-tgrp 40220 df-tendo 40232 df-edring 40234 df-dveca 40480 df-disoa 40506 df-dvech 40556 df-dib 40616 df-dic 40650 df-dih 40706 df-doch 40825 df-djh 40872 df-lcdual 41064 df-mapd 41102 df-hvmap 41234 df-hdmap1 41270 df-hdmap 41271 |
This theorem is referenced by: hdmapval2 41309 |
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