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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapglem5 | Structured version Visualization version GIF version |
Description: Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.) |
Ref | Expression |
---|---|
hdmapglem5.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapglem5.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapglem5.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
hdmapglem5.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapglem5.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapglem5.p | ⊢ + = (+g‘𝑈) |
hdmapglem5.m | ⊢ − = (-g‘𝑈) |
hdmapglem5.q | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmapglem5.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmapglem5.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmapglem5.t | ⊢ × = (.r‘𝑅) |
hdmapglem5.z | ⊢ 0 = (0g‘𝑅) |
hdmapglem5.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapglem5.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hdmapglem5.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapglem5.c | ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) |
hdmapglem5.d | ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) |
hdmapglem5.i | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
hdmapglem5.j | ⊢ (𝜑 → 𝐽 ∈ 𝐵) |
Ref | Expression |
---|---|
hdmapglem5 | ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapglem5.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapglem5.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmapglem5.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlmod 38126 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | hdmapglem5.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | 5 | lmodring 19571 | . . . 4 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
8 | hdmapglem5.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
9 | hdmapglem5.g | . . . 4 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
10 | hdmapglem5.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
11 | hdmapglem5.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
12 | eqid 2818 | . . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | eqid 2818 | . . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
14 | eqid 2818 | . . . . . . . . . 10 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
15 | hdmapglem5.e | . . . . . . . . . 10 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
16 | 1, 12, 13, 2, 10, 14, 15, 3 | dvheveccl 38128 | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
17 | 16 | eldifad 3945 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
18 | 17 | snssd 4734 | . . . . . . 7 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
19 | hdmapglem5.o | . . . . . . . 8 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
20 | 1, 2, 10, 19 | dochssv 38371 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉) |
21 | 3, 18, 20 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉) |
22 | hdmapglem5.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) | |
23 | 21, 22 | sseldd 3965 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
24 | hdmapglem5.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) | |
25 | 21, 24 | sseldd 3965 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
26 | 1, 2, 10, 5, 8, 11, 3, 23, 25 | hdmapipcl 38921 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) ∈ 𝐵) |
27 | 1, 2, 5, 8, 9, 3, 26 | hgmapcl 38905 | . . 3 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) ∈ 𝐵) |
28 | hdmapglem5.t | . . . 4 ⊢ × = (.r‘𝑅) | |
29 | eqid 2818 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
30 | 8, 28, 29 | ringlidm 19250 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘((𝑆‘𝐷)‘𝐶)) ∈ 𝐵) → ((1r‘𝑅) × (𝐺‘((𝑆‘𝐷)‘𝐶))) = (𝐺‘((𝑆‘𝐷)‘𝐶))) |
31 | 7, 27, 30 | syl2anc 584 | . 2 ⊢ (𝜑 → ((1r‘𝑅) × (𝐺‘((𝑆‘𝐷)‘𝐶))) = (𝐺‘((𝑆‘𝐷)‘𝐶))) |
32 | hdmapglem5.p | . . 3 ⊢ + = (+g‘𝑈) | |
33 | hdmapglem5.m | . . 3 ⊢ − = (-g‘𝑈) | |
34 | hdmapglem5.q | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
35 | hdmapglem5.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
36 | 8, 29 | ringidcl 19247 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
37 | 7, 36 | syl 17 | . . 3 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
38 | 1, 2, 5, 29, 9, 3 | hgmapval1 38909 | . . . . 5 ⊢ (𝜑 → (𝐺‘(1r‘𝑅)) = (1r‘𝑅)) |
39 | 38 | oveq2d 7161 | . . . 4 ⊢ (𝜑 → (((𝑆‘𝐷)‘𝐶) × (𝐺‘(1r‘𝑅))) = (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅))) |
40 | 8, 28, 29 | ringridm 19251 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((𝑆‘𝐷)‘𝐶) ∈ 𝐵) → (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅)) = ((𝑆‘𝐷)‘𝐶)) |
41 | 7, 26, 40 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅)) = ((𝑆‘𝐷)‘𝐶)) |
42 | 39, 41 | eqtrd 2853 | . . 3 ⊢ (𝜑 → (((𝑆‘𝐷)‘𝐶) × (𝐺‘(1r‘𝑅))) = ((𝑆‘𝐷)‘𝐶)) |
43 | 1, 15, 19, 2, 10, 32, 33, 34, 5, 8, 28, 35, 11, 9, 3, 22, 24, 26, 37, 42 | hdmapinvlem4 38937 | . 2 ⊢ (𝜑 → ((1r‘𝑅) × (𝐺‘((𝑆‘𝐷)‘𝐶))) = ((𝑆‘𝐶)‘𝐷)) |
44 | 31, 43 | eqtr3d 2855 | 1 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {csn 4557 〈cop 4563 I cid 5452 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 -gcsg 18043 1rcur 19180 Ringcrg 19226 LModclmod 19563 HLchlt 36366 LHypclh 37000 LTrncltrn 37117 DVecHcdvh 38094 ocHcoch 38363 HDMapchdma 38808 HGMapchg 38899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-lshyp 35993 df-lcv 36035 df-lfl 36074 df-lkr 36102 df-ldual 36140 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tgrp 37759 df-tendo 37771 df-edring 37773 df-dveca 38019 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 df-djh 38411 df-lcdual 38603 df-mapd 38641 df-hvmap 38773 df-hdmap1 38809 df-hdmap 38810 df-hgmap 38900 |
This theorem is referenced by: hgmapvvlem1 38939 hdmapglem7 38945 |
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