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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 41111 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 5 | hdmapglem5.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
| 6 | 5 | lmodring 20781 | . . . 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 2730 | . . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 13 | eqid 2730 | . . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
| 14 | eqid 2730 | . . . . . . . . . 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 41113 | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
| 17 | 16 | eldifad 3929 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
| 18 | 17 | snssd 4776 | . . . . . . 7 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
| 19 | hdmapglem5.o | . . . . . . . 8 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
| 20 | 1, 2, 10, 19 | dochssv 41356 | . . . . . . 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 3950 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
| 24 | hdmapglem5.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) | |
| 25 | 21, 24 | sseldd 3950 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
| 26 | 1, 2, 10, 5, 8, 11, 3, 23, 25 | hdmapipcl 41906 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) ∈ 𝐵) |
| 27 | 1, 2, 5, 8, 9, 3, 26 | hgmapcl 41890 | . . 3 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) ∈ 𝐵) |
| 28 | hdmapglem5.t | . . . 4 ⊢ × = (.r‘𝑅) | |
| 29 | eqid 2730 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 30 | 8, 28, 29 | ringlidm 20185 | . . 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 20181 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
| 37 | 7, 36 | syl 17 | . . 3 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
| 38 | 1, 2, 5, 29, 9, 3 | hgmapval1 41894 | . . . . 5 ⊢ (𝜑 → (𝐺‘(1r‘𝑅)) = (1r‘𝑅)) |
| 39 | 38 | oveq2d 7406 | . . . 4 ⊢ (𝜑 → (((𝑆‘𝐷)‘𝐶) × (𝐺‘(1r‘𝑅))) = (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅))) |
| 40 | 8, 28, 29 | ringridm 20186 | . . . . 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 2765 | . . 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 41922 | . 2 ⊢ (𝜑 → ((1r‘𝑅) × (𝐺‘((𝑆‘𝐷)‘𝐶))) = ((𝑆‘𝐶)‘𝐷)) |
| 44 | 31, 43 | eqtr3d 2767 | 1 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3917 {csn 4592 ⟨cop 4598 I cid 5535 ↾ cres 5643 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +gcplusg 17227 .rcmulr 17228 Scalarcsca 17230 ·𝑠 cvsca 17231 0gc0g 17409 -gcsg 18874 1rcur 20097 Ringcrg 20149 LModclmod 20773 HLchlt 39350 LHypclh 39985 LTrncltrn 40102 DVecHcdvh 41079 ocHcoch 41348 HDMapchdma 41793 HGMapchg 41884 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953 |
| 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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-ot 4601 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-mre 17554 df-mrc 17555 df-acs 17557 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-oppg 19285 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-lshyp 38977 df-lcv 39019 df-lfl 39058 df-lkr 39086 df-ldual 39124 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tgrp 40744 df-tendo 40756 df-edring 40758 df-dveca 41004 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349 df-djh 41396 df-lcdual 41588 df-mapd 41626 df-hvmap 41758 df-hdmap1 41794 df-hdmap 41795 df-hgmap 41885 |
| This theorem is referenced by: hgmapvvlem1 41924 hdmapglem7 41930 |
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