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Mathbox for Norm Megill |
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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 4642 | . . . . 5 ⊢ (𝑇 = (0g‘𝑈) → {𝑇} = {(0g‘𝑈)}) | |
2 | 1 | fveq2d 6904 | . . . 4 ⊢ (𝑇 = (0g‘𝑈) → (𝑁‘{𝑇}) = (𝑁‘{(0g‘𝑈)})) |
3 | 2 | fveq2d 6904 | . . 3 ⊢ (𝑇 = (0g‘𝑈) → (𝑀‘(𝑁‘{𝑇})) = (𝑀‘(𝑁‘{(0g‘𝑈)}))) |
4 | fveq2 6900 | . . . . 5 ⊢ (𝑇 = (0g‘𝑈) → (𝑆‘𝑇) = (𝑆‘(0g‘𝑈))) | |
5 | 4 | sneqd 4644 | . . . 4 ⊢ (𝑇 = (0g‘𝑈) → {(𝑆‘𝑇)} = {(𝑆‘(0g‘𝑈))}) |
6 | 5 | fveq2d 6904 | . . 3 ⊢ (𝑇 = (0g‘𝑈) → (𝐿‘{(𝑆‘𝑇)}) = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
7 | 3, 6 | eqeq12d 2741 | . 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 479 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
18 | eqid 2725 | . . 3 ⊢ 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
19 | eqid 2725 | . . 3 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
20 | eqid 2725 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
21 | eqid 2725 | . . 3 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊) | |
22 | eqid 2725 | . . 3 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊) | |
23 | hdmap10.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
24 | 23 | anim1i 613 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ (0g‘𝑈))) |
25 | eldifsn 4794 | . . . 4 ⊢ (𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)}) ↔ (𝑇 ∈ 𝑉 ∧ 𝑇 ≠ (0g‘𝑈))) | |
26 | 24, 25 | sylibr 233 | . . 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 41486 | . 2 ⊢ ((𝜑 ∧ 𝑇 ≠ (0g‘𝑈)) → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
28 | 8, 9, 16 | dvhlmod 40757 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
29 | 19, 11 | lspsn0 20932 | . . . . 5 ⊢ (𝑈 ∈ LMod → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
30 | 28, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁‘{(0g‘𝑈)}) = {(0g‘𝑈)}) |
31 | 30 | fveq2d 6904 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(0g‘𝑈)})) = (𝑀‘{(0g‘𝑈)})) |
32 | eqid 2725 | . . . 4 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
33 | 8, 14, 9, 19, 12, 32, 16 | mapd0 41312 | . . 3 ⊢ (𝜑 → (𝑀‘{(0g‘𝑈)}) = {(0g‘𝐶)}) |
34 | 8, 9, 19, 12, 32, 15, 16 | hdmapval0 41480 | . . . . . 6 ⊢ (𝜑 → (𝑆‘(0g‘𝑈)) = (0g‘𝐶)) |
35 | 34 | sneqd 4644 | . . . . 5 ⊢ (𝜑 → {(𝑆‘(0g‘𝑈))} = {(0g‘𝐶)}) |
36 | 35 | fveq2d 6904 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘(0g‘𝑈))}) = (𝐿‘{(0g‘𝐶)})) |
37 | 8, 12, 16 | lcdlmod 41239 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod) |
38 | 32, 13 | lspsn0 20932 | . . . . 5 ⊢ (𝐶 ∈ LMod → (𝐿‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
39 | 37, 38 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐿‘{(0g‘𝐶)}) = {(0g‘𝐶)}) |
40 | 36, 39 | eqtr2d 2766 | . . 3 ⊢ (𝜑 → {(0g‘𝐶)} = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
41 | 31, 33, 40 | 3eqtrd 2769 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(0g‘𝑈)})) = (𝐿‘{(𝑆‘(0g‘𝑈))})) |
42 | 7, 27, 41 | pm2.61ne 3016 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3943 {csn 4632 〈cop 4638 I cid 5578 ↾ cres 5683 ‘cfv 6553 Basecbs 17208 0gc0g 17449 LModclmod 20783 LSpanclspn 20895 HLchlt 38996 LHypclh 39631 LTrncltrn 39748 DVecHcdvh 40725 LCDualclcd 41233 mapdcmpd 41271 HVMapchvm 41403 HDMap1chdma1 41438 HDMapchdma 41439 |
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 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-riotaBAD 38599 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-tpos 8240 df-undef 8287 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-0g 17451 df-mre 17594 df-mrc 17595 df-acs 17597 df-proset 18315 df-poset 18333 df-plt 18350 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-p0 18445 df-p1 18446 df-lat 18452 df-clat 18519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19112 df-cntz 19306 df-oppg 19335 df-lsm 19629 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-oppr 20311 df-dvdsr 20334 df-unit 20335 df-invr 20365 df-dvr 20378 df-drng 20666 df-lmod 20785 df-lss 20856 df-lsp 20896 df-lvec 21028 df-lsatoms 38622 df-lshyp 38623 df-lcv 38665 df-lfl 38704 df-lkr 38732 df-ldual 38770 df-oposet 38822 df-ol 38824 df-oml 38825 df-covers 38912 df-ats 38913 df-atl 38944 df-cvlat 38968 df-hlat 38997 df-llines 39145 df-lplanes 39146 df-lvols 39147 df-lines 39148 df-psubsp 39150 df-pmap 39151 df-padd 39443 df-lhyp 39635 df-laut 39636 df-ldil 39751 df-ltrn 39752 df-trl 39806 df-tgrp 40390 df-tendo 40402 df-edring 40404 df-dveca 40650 df-disoa 40676 df-dvech 40726 df-dib 40786 df-dic 40820 df-dih 40876 df-doch 40995 df-djh 41042 df-lcdual 41234 df-mapd 41272 df-hvmap 41404 df-hdmap1 41440 df-hdmap 41441 |
This theorem is referenced by: hdmapeq0 41491 hdmaprnlem1N 41496 hdmaprnlem3uN 41498 hdmaprnlem6N 41501 hdmaprnlem8N 41503 hdmaprnlem3eN 41505 hdmap14lem1a 41513 hdmap14lem9 41523 hgmaprnlem2N 41544 hdmaplkr 41560 |
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