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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 41609 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 5 | hdmapglem5.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
| 6 | 5 | lmodring 20865 | . . . 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 2740 | . . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 13 | eqid 2740 | . . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
| 14 | eqid 2740 | . . . . . . . . . 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 41611 | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
| 17 | 16 | eldifad 3902 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
| 18 | 17 | snssd 4725 | . . . . . . 7 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
| 19 | hdmapglem5.o | . . . . . . . 8 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
| 20 | 1, 2, 10, 19 | dochssv 41854 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉) |
| 21 | 3, 18, 20 | syl2anc 590 | . . . . . 6 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉) |
| 22 | hdmapglem5.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) | |
| 23 | 21, 22 | sseldd 3923 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
| 24 | hdmapglem5.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) | |
| 25 | 21, 24 | sseldd 3923 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
| 26 | 1, 2, 10, 5, 8, 11, 3, 23, 25 | hdmapipcl 42404 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) ∈ 𝐵) |
| 27 | 1, 2, 5, 8, 9, 3, 26 | hgmapcl 42388 | . . 3 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) ∈ 𝐵) |
| 28 | hdmapglem5.t | . . . 4 ⊢ × = (.r‘𝑅) | |
| 29 | eqid 2740 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 30 | 8, 28, 29 | ringlidm 20248 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘((𝑆‘𝐷)‘𝐶)) ∈ 𝐵) → ((1r‘𝑅) × (𝐺‘((𝑆‘𝐷)‘𝐶))) = (𝐺‘((𝑆‘𝐷)‘𝐶))) |
| 31 | 7, 27, 30 | syl2anc 590 | . 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 20244 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
| 37 | 7, 36 | syl 17 | . . 3 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
| 38 | 1, 2, 5, 29, 9, 3 | hgmapval1 42392 | . . . . 5 ⊢ (𝜑 → (𝐺‘(1r‘𝑅)) = (1r‘𝑅)) |
| 39 | 38 | oveq2d 7379 | . . . 4 ⊢ (𝜑 → (((𝑆‘𝐷)‘𝐶) × (𝐺‘(1r‘𝑅))) = (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅))) |
| 40 | 8, 28, 29 | ringridm 20249 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((𝑆‘𝐷)‘𝐶) ∈ 𝐵) → (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅)) = ((𝑆‘𝐷)‘𝐶)) |
| 41 | 7, 26, 40 | syl2anc 590 | . . . 4 ⊢ (𝜑 → (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅)) = ((𝑆‘𝐷)‘𝐶)) |
| 42 | 39, 41 | eqtrd 2775 | . . 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 42420 | . 2 ⊢ (𝜑 → ((1r‘𝑅) × (𝐺‘((𝑆‘𝐷)‘𝐶))) = ((𝑆‘𝐶)‘𝐷)) |
| 44 | 31, 43 | eqtr3d 2777 | 1 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ⊆ wss 3890 {csn 4562 ⟨cop 4568 I cid 5519 ↾ cres 5627 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 +gcplusg 17218 .rcmulr 17219 Scalarcsca 17221 ·𝑠 cvsca 17222 0gc0g 17400 -gcsg 18909 1rcur 20160 Ringcrg 20212 LModclmod 20857 HLchlt 39849 LHypclh 40483 LTrncltrn 40600 DVecHcdvh 41577 ocHcoch 41846 HDMapchdma 42291 HGMapchg 42382 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39452 |
| This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-ot 4571 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-mre 17546 df-mrc 17547 df-acs 17549 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-oppg 19319 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-nzr 20492 df-rlreg 20673 df-domn 20674 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39475 df-lshyp 39476 df-lcv 39518 df-lfl 39557 df-lkr 39585 df-ldual 39623 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658 df-tgrp 41242 df-tendo 41254 df-edring 41256 df-dveca 41502 df-disoa 41528 df-dvech 41578 df-dib 41638 df-dic 41672 df-dih 41728 df-doch 41847 df-djh 41894 df-lcdual 42086 df-mapd 42124 df-hvmap 42256 df-hdmap1 42292 df-hdmap 42293 df-hgmap 42383 |
| This theorem is referenced by: hgmapvvlem1 42422 hdmapglem7 42428 |
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