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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapvvlem1 | Structured version Visualization version GIF version |
Description: Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our 𝐸, 𝐶, 𝐷, 𝑌, 𝑋 correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.) |
Ref | Expression |
---|---|
hdmapglem6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapglem6.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapglem6.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
hdmapglem6.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapglem6.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapglem6.q | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmapglem6.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmapglem6.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmapglem6.t | ⊢ × = (.r‘𝑅) |
hdmapglem6.z | ⊢ 0 = (0g‘𝑅) |
hdmapglem6.i | ⊢ 1 = (1r‘𝑅) |
hdmapglem6.n | ⊢ 𝑁 = (invr‘𝑅) |
hdmapglem6.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapglem6.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hdmapglem6.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapglem6.x | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) |
hdmapglem6.c | ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) |
hdmapglem6.d | ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) |
hdmapglem6.cd | ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 ) |
hdmapglem6.y | ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 })) |
hdmapglem6.yx | ⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = 1 ) |
Ref | Expression |
---|---|
hgmapvvlem1 | ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapglem6.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapglem6.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmapglem6.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlmod 40520 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | hdmapglem6.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | 5 | lmodring 20740 | . . . . 5 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
8 | hdmapglem6.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
9 | hdmapglem6.g | . . . . 5 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
10 | hdmapglem6.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) | |
11 | 10 | eldifad 3956 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
12 | 1, 2, 5, 8, 9, 3, 11 | hgmapcl 41299 | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵) |
13 | 1, 2, 5, 8, 9, 3, 12 | hgmapcl 41299 | . . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) |
14 | hdmapglem6.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 })) | |
15 | 14 | eldifad 3956 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
16 | 1, 2, 5, 8, 9, 3, 15 | hgmapcl 41299 | . . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵) |
17 | 1, 2, 3 | dvhlvec 40519 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
18 | 5 | lvecdrng 20979 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing) |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) |
20 | eldifsni 4789 | . . . . . . 7 ⊢ (𝑌 ∈ (𝐵 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
21 | 14, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
22 | hdmapglem6.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
23 | 1, 2, 5, 8, 22, 9, 3, 15 | hgmapeq0 41314 | . . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 )) |
24 | 23 | necon3bid 2980 | . . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑌) ≠ 0 ↔ 𝑌 ≠ 0 )) |
25 | 21, 24 | mpbird 257 | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑌) ≠ 0 ) |
26 | hdmapglem6.n | . . . . . 6 ⊢ 𝑁 = (invr‘𝑅) | |
27 | 8, 22, 26 | drnginvrcl 20635 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
28 | 19, 16, 25, 27 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
29 | hdmapglem6.t | . . . . 5 ⊢ × = (.r‘𝑅) | |
30 | 8, 29 | ringass 20184 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((𝐺‘(𝐺‘𝑋)) ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)) → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
31 | 7, 13, 16, 28, 30 | syl13anc 1370 | . . 3 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
32 | hdmapglem6.i | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
33 | 8, 22, 29, 32, 26 | drnginvrr 20639 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
34 | 19, 16, 25, 33 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
35 | 34 | oveq2d 7430 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 )) |
36 | 8, 29, 32 | ringridm 20195 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
37 | 7, 13, 36 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
38 | 31, 35, 37 | 3eqtrrd 2772 | . 2 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌)))) |
39 | hdmapglem6.yx | . . . . . . 7 ⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = 1 ) | |
40 | 39 | fveq2d 6895 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 )) |
41 | 1, 2, 5, 8, 29, 9, 3, 15, 12 | hgmapmul 41305 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
42 | 40, 41 | eqtr3d 2769 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
43 | hdmapglem6.cd | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 ) | |
44 | 43 | fveq2d 6895 | . . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = (𝐺‘ 1 )) |
45 | hdmapglem6.e | . . . . . . 7 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
46 | hdmapglem6.o | . . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
47 | hdmapglem6.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
48 | eqid 2727 | . . . . . . 7 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
49 | eqid 2727 | . . . . . . 7 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
50 | hdmapglem6.q | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑈) | |
51 | hdmapglem6.s | . . . . . . 7 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
52 | hdmapglem6.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) | |
53 | hdmapglem6.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) | |
54 | 1, 45, 46, 2, 47, 48, 49, 50, 5, 8, 29, 22, 51, 9, 3, 52, 53, 15, 11 | hdmapglem5 41332 | . . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
55 | 44, 54 | eqtr3d 2769 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷)) |
56 | 42, 55 | eqtr3d 2769 | . . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
57 | 39, 43 | eqtr4d 2770 | . . . . 5 ⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = ((𝑆‘𝐷)‘𝐶)) |
58 | 1, 45, 46, 2, 47, 48, 49, 50, 5, 8, 29, 22, 51, 9, 3, 52, 53, 15, 11, 57 | hdmapinvlem4 41331 | . . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
59 | 56, 58 | eqtr4d 2770 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌))) |
60 | 59 | oveq1d 7429 | . 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌)))) |
61 | 8, 29 | ringass 20184 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝑁‘(𝐺‘𝑌)) ∈ 𝐵)) → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
62 | 7, 11, 16, 28, 61 | syl13anc 1370 | . . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
63 | 34 | oveq2d 7430 | . . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 )) |
64 | 8, 29, 32 | ringridm 20195 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋) |
65 | 7, 11, 64 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋) |
66 | 62, 63, 65 | 3eqtrd 2771 | . 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋) |
67 | 38, 60, 66 | 3eqtrd 2771 | 1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∖ cdif 3941 {csn 4624 〈cop 4630 I cid 5569 ↾ cres 5674 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 +gcplusg 17224 .rcmulr 17225 Scalarcsca 17227 ·𝑠 cvsca 17228 0gc0g 17412 -gcsg 18883 1rcur 20112 Ringcrg 20164 invrcinvr 20315 DivRingcdr 20613 LModclmod 20732 LVecclvec 20976 HLchlt 38759 LHypclh 39394 LTrncltrn 39511 DVecHcdvh 40488 ocHcoch 40757 HDMapchdma 41202 HGMapchg 41293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-riotaBAD 38362 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-0g 17414 df-mre 17557 df-mrc 17558 df-acs 17560 df-proset 18278 df-poset 18296 df-plt 18313 df-lub 18329 df-glb 18330 df-join 18331 df-meet 18332 df-p0 18408 df-p1 18409 df-lat 18415 df-clat 18482 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-cntz 19259 df-oppg 19288 df-lsm 19582 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-drng 20615 df-lmod 20734 df-lss 20805 df-lsp 20845 df-lvec 20977 df-lsatoms 38385 df-lshyp 38386 df-lcv 38428 df-lfl 38467 df-lkr 38495 df-ldual 38533 df-oposet 38585 df-ol 38587 df-oml 38588 df-covers 38675 df-ats 38676 df-atl 38707 df-cvlat 38731 df-hlat 38760 df-llines 38908 df-lplanes 38909 df-lvols 38910 df-lines 38911 df-psubsp 38913 df-pmap 38914 df-padd 39206 df-lhyp 39398 df-laut 39399 df-ldil 39514 df-ltrn 39515 df-trl 39569 df-tgrp 40153 df-tendo 40165 df-edring 40167 df-dveca 40413 df-disoa 40439 df-dvech 40489 df-dib 40549 df-dic 40583 df-dih 40639 df-doch 40758 df-djh 40805 df-lcdual 40997 df-mapd 41035 df-hvmap 41167 df-hdmap1 41203 df-hdmap 41204 df-hgmap 41294 |
This theorem is referenced by: hgmapvvlem2 41334 |
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