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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 42261 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 5 | hdmapglem5.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
| 6 | 5 | lmodring 21149 | . . . 4 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring) |
| 7 | 4, 6 | syl 18 | . . 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 2760 | . . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 13 | eqid 2760 | . . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
| 14 | eqid 2760 | . . . . . . . . . 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 42263 | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
| 17 | 16 | eldifad 4051 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
| 18 | 17 | snssd 4743 | . . . . . . 7 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) |
| 19 | hdmapglem5.o | . . . . . . . 8 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
| 20 | 1, 2, 10, 19 | dochssv 42506 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉) |
| 21 | 3, 18, 20 | syl2anc 596 | . . . . . 6 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉) |
| 22 | hdmapglem5.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) | |
| 23 | 21, 22 | sseldd 3909 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
| 24 | hdmapglem5.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) | |
| 25 | 21, 24 | sseldd 3909 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
| 26 | 1, 2, 10, 5, 8, 11, 3, 23, 25 | hdmapipcl 43056 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) ∈ 𝐵) |
| 27 | 1, 2, 5, 8, 9, 3, 26 | hgmapcl 43040 | . . 3 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) ∈ 𝐵) |
| 28 | hdmapglem5.t | . . . 4 ⊢ × = (.r‘𝑅) | |
| 29 | eqid 2760 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 30 | 8, 28, 29 | ringlidm 20504 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘((𝑆‘𝐷)‘𝐶)) ∈ 𝐵) → ((1r‘𝑅) × (𝐺‘((𝑆‘𝐷)‘𝐶))) = (𝐺‘((𝑆‘𝐷)‘𝐶))) |
| 31 | 7, 27, 30 | syl2anc 596 | . 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 20500 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
| 37 | 7, 36 | syl 18 | . . 3 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
| 38 | 1, 2, 5, 29, 9, 3 | hgmapval1 43044 | . . . . 5 ⊢ (𝜑 → (𝐺‘(1r‘𝑅)) = (1r‘𝑅)) |
| 39 | 38 | oveq2d 7437 | . . . 4 ⊢ (𝜑 → (((𝑆‘𝐷)‘𝐶) × (𝐺‘(1r‘𝑅))) = (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅))) |
| 40 | 8, 28, 29 | ringridm 20505 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((𝑆‘𝐷)‘𝐶) ∈ 𝐵) → (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅)) = ((𝑆‘𝐷)‘𝐶)) |
| 41 | 7, 26, 40 | syl2anc 596 | . . . 4 ⊢ (𝜑 → (((𝑆‘𝐷)‘𝐶) × (1r‘𝑅)) = ((𝑆‘𝐷)‘𝐶)) |
| 42 | 39, 41 | eqtrd 2795 | . . 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 43072 | . 2 ⊢ (𝜑 → ((1r‘𝑅) × (𝐺‘((𝑆‘𝐷)‘𝐶))) = ((𝑆‘𝐶)‘𝐷)) |
| 44 | 31, 43 | eqtr3d 2797 | 1 ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 {csn 4584 ⟨cop 4834 I cid 5544 ↾ cres 5652 ‘cfv 6539 (class class class)co 7421 Basecbs 17393 +gcplusg 17434 .rcmulr 17435 Scalarcsca 17437 ·𝑠 cvsca 17438 0gc0g 17616 -gcsg 19152 1rcur 20413 Ringcrg 20465 LModclmod 21141 HLchlt 40501 LHypclh 41135 LTrncltrn 41252 DVecHcdvh 42229 ocHcoch 42498 HDMapchdma 42943 HGMapchg 43034 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-riotaBAD 40104 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8247 df-undef 8294 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-n0 12611 df-z 12698 df-uz 12970 df-fz 13644 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-sca 17450 df-vsca 17451 df-0g 17618 df-mre 17762 df-mrc 17763 df-acs 17765 df-proset 18474 df-poset 18493 df-plt 18508 df-lub 18524 df-glb 18525 df-join 18526 df-meet 18527 df-p0 18603 df-p1 18604 df-lat 18612 df-clat 18679 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-grp 19153 df-minusg 19154 df-sbg 19155 df-subg 19339 df-cntz 19537 df-oppg 19566 df-lsm 19856 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-oppr 20573 df-dvdsr 20593 df-unit 20594 df-invr 20624 df-dvr 20637 df-nzr 20769 df-rlreg 20952 df-domn 20953 df-drng 20988 df-lmod 21143 df-lss 21213 df-lsp 21253 df-lvec 21384 df-lsatoms 40127 df-lshyp 40128 df-lcv 40170 df-lfl 40209 df-lkr 40237 df-ldual 40275 df-oposet 40327 df-ol 40329 df-oml 40330 df-covers 40417 df-ats 40418 df-atl 40449 df-cvlat 40473 df-hlat 40502 df-llines 40649 df-lplanes 40650 df-lvols 40651 df-lines 40652 df-psubsp 40654 df-pmap 40655 df-padd 40947 df-lhyp 41139 df-laut 41140 df-ldil 41255 df-ltrn 41256 df-trl 41310 df-tgrp 41894 df-tendo 41906 df-edring 41908 df-dveca 42154 df-disoa 42180 df-dvech 42230 df-dib 42290 df-dic 42324 df-dih 42380 df-doch 42499 df-djh 42546 df-lcdual 42738 df-mapd 42776 df-hvmap 42908 df-hdmap1 42944 df-hdmap 42945 df-hgmap 43035 |
| This theorem is used by: hgmapvvlem1 43074 hdmapglem7 43080 |
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