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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap10 | Structured version Visualization version GIF version |
Description: Part 10 in [Baer] p. 48 line 33, (Ft)* = G(tS) in their notation (S = sigma). (Contributed by NM, 17-May-2015.) |
Ref | Expression |
---|---|
hdmap10.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap10.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap10.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap10.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap10.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap10.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap10.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap10.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap10.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap10.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmap10 | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4535 | . . . . 5 ⊢ (𝑇 = (0g‘𝑈) → {𝑇} = {(0g‘𝑈)}) | |
2 | 1 | fveq2d 6649 | . . . 4 ⊢ (𝑇 = (0g‘𝑈) → (𝑁‘{𝑇}) = (𝑁‘{(0g‘𝑈)})) |
3 | 2 | fveq2d 6649 | . . 3 ⊢ (𝑇 = (0g‘𝑈) → (𝑀‘(𝑁‘{𝑇})) = (𝑀‘(𝑁‘{(0g‘𝑈)}))) |
4 | fveq2 6645 | . . . . 5 ⊢ (𝑇 = (0g‘𝑈) → (𝑆‘𝑇) = (𝑆‘(0g‘𝑈))) | |
5 | 4 | sneqd 4537 | . . . 4 ⊢ (𝑇 = (0g‘𝑈) → {(𝑆‘𝑇)} = {(𝑆‘(0g‘𝑈))}) |
6 | 5 | fveq2d 6649 | . . 3 ⊢ (𝑇 = (0g‘𝑈) → (𝐿‘{(𝑆‘𝑇)}) = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
7 | 3, 6 | eqeq12d 2814 | . 2 ⊢ (𝑇 = (0g‘𝑈) → ((𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)}) ↔ (𝑀‘(𝑁‘{(0g‘𝑈)})) = (𝐿‘{(𝑆‘(0g‘𝑈))}))) |
8 | hdmap10.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | hdmap10.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | hdmap10.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
11 | hdmap10.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
12 | hdmap10.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
13 | hdmap10.l | . . 3 ⊢ 𝐿 = (LSpan‘𝐶) | |
14 | hdmap10.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
15 | hdmap10.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
16 | hdmap10.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 16 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
18 | eqid 2798 | . . 3 ⊢ 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
19 | eqid 2798 | . . 3 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
20 | eqid 2798 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
21 | eqid 2798 | . . 3 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊) | |
22 | eqid 2798 | . . 3 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊) | |
23 | hdmap10.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
24 | 23 | anim1i 617 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ (0g‘𝑈))) |
25 | eldifsn 4680 | . . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)}) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ (0g‘𝑈))) | |
26 | 24, 25 | sylibr 237 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
27 | 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 26 | hdmap10lem 39135 | . 2 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
28 | 8, 9, 16 | dvhlmod 38406 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
29 | 19, 11 | lspsn0 19773 | . . . . 5 ⊢ (𝑈 ∈ LMod → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
30 | 28, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
31 | 30 | fveq2d 6649 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(0g‘𝑈)})) = (𝑀‘{(0g‘𝑈)})) |
32 | eqid 2798 | . . . 4 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
33 | 8, 14, 9, 19, 12, 32, 16 | mapd0 38961 | . . 3 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
34 | 8, 9, 19, 12, 32, 15, 16 | hdmapval0 39129 | . . . . . 6 ⊢ (𝜑 → (𝑆‘(0g‘𝑈)) = (0g‘𝐶)) |
35 | 34 | sneqd 4537 | . . . . 5 ⊢ (𝜑 → {(𝑆‘(0g‘𝑈))} = {(0g‘𝐶)}) |
36 | 35 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘(0g‘𝑈))}) = (𝐿‘{(0g‘𝐶)})) |
37 | 8, 12, 16 | lcdlmod 38888 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod) |
38 | 32, 13 | lspsn0 19773 | . . . . 5 ⊢ (𝐶 ∈ LMod → (𝐿‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
39 | 37, 38 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐿‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
40 | 36, 39 | eqtr2d 2834 | . . 3 ⊢ (𝜑 → {(0g‘𝐶)} = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
41 | 31, 33, 40 | 3eqtrd 2837 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(0g‘𝑈)})) = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
42 | 7, 27, 41 | pm2.61ne 3072 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 {csn 4525 〈cop 4531 I cid 5424 ↾ cres 5521 ‘cfv 6324 Basecbs 16475 0gc0g 16705 LModclmod 19627 LSpanclspn 19736 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 DVecHcdvh 38374 LCDualclcd 38882 mapdcmpd 38920 HVMapchvm 39052 HDMap1chdma1 39087 HDMapchdma 39088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lshyp 36273 df-lcv 36315 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 df-lcdual 38883 df-mapd 38921 df-hvmap 39053 df-hdmap1 39089 df-hdmap 39090 |
This theorem is referenced by: hdmapeq0 39140 hdmaprnlem1N 39145 hdmaprnlem3uN 39147 hdmaprnlem6N 39150 hdmaprnlem8N 39152 hdmaprnlem3eN 39154 hdmap14lem1a 39162 hdmap14lem9 39172 hgmaprnlem2N 39193 hdmaplkr 39209 |
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