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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 41556 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 5 | hdmapglem6.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈) | |
| 6 | 5 | lmodring 20863 | . . . . 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 3901 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 12 | 1, 2, 5, 8, 9, 3, 11 | hgmapcl 42335 | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵) |
| 13 | 1, 2, 5, 8, 9, 3, 12 | hgmapcl 42335 | . . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) |
| 14 | hdmapglem6.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 })) | |
| 15 | 14 | eldifad 3901 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| 16 | 1, 2, 5, 8, 9, 3, 15 | hgmapcl 42335 | . . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵) |
| 17 | 1, 2, 3 | dvhlvec 41555 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
| 18 | 5 | lvecdrng 21100 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing) |
| 19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) |
| 20 | eldifsni 4735 | . . . . . . 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 42350 | . . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌) = 0 ↔ 𝑌 = 0 )) |
| 24 | 23 | necon3bid 2976 | . . . . . 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 20730 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
| 28 | 19, 16, 25, 27 | syl3anc 1374 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
| 29 | hdmapglem6.t | . . . . 5 ⊢ × = (.r‘𝑅) | |
| 30 | 8, 29 | ringass 20234 | . . . 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 1375 | . . 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 20734 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
| 34 | 19, 16, 25, 33 | syl3anc 1374 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
| 35 | 34 | oveq2d 7383 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 )) |
| 36 | 8, 29, 32 | ringridm 20251 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
| 37 | 7, 13, 36 | syl2anc 585 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
| 38 | 31, 35, 37 | 3eqtrrd 2776 | . 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 6844 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 )) |
| 41 | 1, 2, 5, 8, 29, 9, 3, 15, 12 | hgmapmul 42341 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
| 42 | 40, 41 | eqtr3d 2773 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
| 43 | hdmapglem6.cd | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 ) | |
| 44 | 43 | fveq2d 6844 | . . . . . 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 2736 | . . . . . . 7 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
| 49 | eqid 2736 | . . . . . . 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 42368 | . . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
| 55 | 44, 54 | eqtr3d 2773 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷)) |
| 56 | 42, 55 | eqtr3d 2773 | . . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
| 57 | 39, 43 | eqtr4d 2774 | . . . . 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 42367 | . . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
| 59 | 56, 58 | eqtr4d 2774 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌))) |
| 60 | 59 | oveq1d 7382 | . 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌)))) |
| 61 | 8, 29 | ringass 20234 | . . . 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 1375 | . . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
| 63 | 34 | oveq2d 7383 | . . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 )) |
| 64 | 8, 29, 32 | ringridm 20251 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋) |
| 65 | 7, 11, 64 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋) |
| 66 | 62, 63, 65 | 3eqtrd 2775 | . 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋) |
| 67 | 38, 60, 66 | 3eqtrd 2775 | 1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∖ cdif 3886 {csn 4567 ⟨cop 4573 I cid 5525 ↾ cres 5633 ‘cfv 6498 (class class class)co 7367 Basecbs 17179 +gcplusg 17220 .rcmulr 17221 Scalarcsca 17223 ·𝑠 cvsca 17224 0gc0g 17402 -gcsg 18911 1rcur 20162 Ringcrg 20214 invrcinvr 20367 DivRingcdr 20706 LModclmod 20855 LVecclvec 21097 HLchlt 39796 LHypclh 40430 LTrncltrn 40547 DVecHcdvh 41524 ocHcoch 41793 HDMapchdma 42238 HGMapchg 42329 |
| 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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39399 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-undef 8223 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17548 df-mrc 17549 df-acs 17551 df-proset 18260 df-poset 18279 df-plt 18294 df-lub 18310 df-glb 18311 df-join 18312 df-meet 18313 df-p0 18389 df-p1 18390 df-lat 18398 df-clat 18465 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-cntz 19292 df-oppg 19321 df-lsm 19611 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-nzr 20490 df-rlreg 20671 df-domn 20672 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lvec 21098 df-lsatoms 39422 df-lshyp 39423 df-lcv 39465 df-lfl 39504 df-lkr 39532 df-ldual 39570 df-oposet 39622 df-ol 39624 df-oml 39625 df-covers 39712 df-ats 39713 df-atl 39744 df-cvlat 39768 df-hlat 39797 df-llines 39944 df-lplanes 39945 df-lvols 39946 df-lines 39947 df-psubsp 39949 df-pmap 39950 df-padd 40242 df-lhyp 40434 df-laut 40435 df-ldil 40550 df-ltrn 40551 df-trl 40605 df-tgrp 41189 df-tendo 41201 df-edring 41203 df-dveca 41449 df-disoa 41475 df-dvech 41525 df-dib 41585 df-dic 41619 df-dih 41675 df-doch 41794 df-djh 41841 df-lcdual 42033 df-mapd 42071 df-hvmap 42203 df-hdmap1 42239 df-hdmap 42240 df-hgmap 42330 |
| This theorem is referenced by: hgmapvvlem2 42370 |
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