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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 41129 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 5 | hdmapglem6.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈) | |
| 6 | 5 | lmodring 20825 | . . . . 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 3938 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 12 | 1, 2, 5, 8, 9, 3, 11 | hgmapcl 41908 | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐵) |
| 13 | 1, 2, 5, 8, 9, 3, 12 | hgmapcl 41908 | . . . 4 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) |
| 14 | hdmapglem6.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 })) | |
| 15 | 14 | eldifad 3938 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| 16 | 1, 2, 5, 8, 9, 3, 15 | hgmapcl 41908 | . . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐵) |
| 17 | 1, 2, 3 | dvhlvec 41128 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
| 18 | 5 | lvecdrng 21063 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑅 ∈ DivRing) |
| 19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) |
| 20 | eldifsni 4766 | . . . . . . 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 41923 | . . . . . . 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 20713 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
| 28 | 19, 16, 25, 27 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝐺‘𝑌)) ∈ 𝐵) |
| 29 | hdmapglem6.t | . . . . 5 ⊢ × = (.r‘𝑅) | |
| 30 | 8, 29 | ringass 20213 | . . . 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 1374 | . . 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 20717 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (𝐺‘𝑌) ∈ 𝐵 ∧ (𝐺‘𝑌) ≠ 0 ) → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
| 34 | 19, 16, 25, 33 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))) = 1 ) |
| 35 | 34 | oveq2d 7421 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = ((𝐺‘(𝐺‘𝑋)) × 1 )) |
| 36 | 8, 29, 32 | ringridm 20230 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘(𝐺‘𝑋)) ∈ 𝐵) → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
| 37 | 7, 13, 36 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × 1 ) = (𝐺‘(𝐺‘𝑋))) |
| 38 | 31, 35, 37 | 3eqtrrd 2775 | . 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 6880 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = (𝐺‘ 1 )) |
| 41 | 1, 2, 5, 8, 29, 9, 3, 15, 12 | hgmapmul 41914 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑌 × (𝐺‘𝑋))) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
| 42 | 40, 41 | eqtr3d 2772 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌))) |
| 43 | hdmapglem6.cd | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 ) | |
| 44 | 43 | fveq2d 6880 | . . . . . 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 2735 | . . . . . . 7 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
| 49 | eqid 2735 | . . . . . . 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 41941 | . . . . . 6 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
| 55 | 44, 54 | eqtr3d 2772 | . . . . 5 ⊢ (𝜑 → (𝐺‘ 1 ) = ((𝑆‘𝐶)‘𝐷)) |
| 56 | 42, 55 | eqtr3d 2772 | . . . 4 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
| 57 | 39, 43 | eqtr4d 2773 | . . . . 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 41940 | . . . 4 ⊢ (𝜑 → (𝑋 × (𝐺‘𝑌)) = ((𝑆‘𝐶)‘𝐷)) |
| 59 | 56, 58 | eqtr4d 2773 | . . 3 ⊢ (𝜑 → ((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) = (𝑋 × (𝐺‘𝑌))) |
| 60 | 59 | oveq1d 7420 | . 2 ⊢ (𝜑 → (((𝐺‘(𝐺‘𝑋)) × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌)))) |
| 61 | 8, 29 | ringass 20213 | . . . 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 1374 | . . 3 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌))))) |
| 63 | 34 | oveq2d 7421 | . . 3 ⊢ (𝜑 → (𝑋 × ((𝐺‘𝑌) × (𝑁‘(𝐺‘𝑌)))) = (𝑋 × 1 )) |
| 64 | 8, 29, 32 | ringridm 20230 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 × 1 ) = 𝑋) |
| 65 | 7, 11, 64 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑋 × 1 ) = 𝑋) |
| 66 | 62, 63, 65 | 3eqtrd 2774 | . 2 ⊢ (𝜑 → ((𝑋 × (𝐺‘𝑌)) × (𝑁‘(𝐺‘𝑌))) = 𝑋) |
| 67 | 38, 60, 66 | 3eqtrd 2774 | 1 ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∖ cdif 3923 {csn 4601 ⟨cop 4607 I cid 5547 ↾ cres 5656 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 +gcplusg 17271 .rcmulr 17272 Scalarcsca 17274 ·𝑠 cvsca 17275 0gc0g 17453 -gcsg 18918 1rcur 20141 Ringcrg 20193 invrcinvr 20347 DivRingcdr 20689 LModclmod 20817 LVecclvec 21060 HLchlt 39368 LHypclh 40003 LTrncltrn 40120 DVecHcdvh 41097 ocHcoch 41366 HDMapchdma 41811 HGMapchg 41902 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-riotaBAD 38971 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-ot 4610 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-0g 17455 df-mre 17598 df-mrc 17599 df-acs 17601 df-proset 18306 df-poset 18325 df-plt 18340 df-lub 18356 df-glb 18357 df-join 18358 df-meet 18359 df-p0 18435 df-p1 18436 df-lat 18442 df-clat 18509 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cntz 19300 df-oppg 19329 df-lsm 19617 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-nzr 20473 df-rlreg 20654 df-domn 20655 df-drng 20691 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lvec 21061 df-lsatoms 38994 df-lshyp 38995 df-lcv 39037 df-lfl 39076 df-lkr 39104 df-ldual 39142 df-oposet 39194 df-ol 39196 df-oml 39197 df-covers 39284 df-ats 39285 df-atl 39316 df-cvlat 39340 df-hlat 39369 df-llines 39517 df-lplanes 39518 df-lvols 39519 df-lines 39520 df-psubsp 39522 df-pmap 39523 df-padd 39815 df-lhyp 40007 df-laut 40008 df-ldil 40123 df-ltrn 40124 df-trl 40178 df-tgrp 40762 df-tendo 40774 df-edring 40776 df-dveca 41022 df-disoa 41048 df-dvech 41098 df-dib 41158 df-dic 41192 df-dih 41248 df-doch 41367 df-djh 41414 df-lcdual 41606 df-mapd 41644 df-hvmap 41776 df-hdmap1 41812 df-hdmap 41813 df-hgmap 41903 |
| This theorem is referenced by: hgmapvvlem2 41943 |
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