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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmapvs | Structured version Visualization version GIF version |
Description: Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.) |
Ref | Expression |
---|---|
hgmapvs.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hgmapvs.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hgmapvs.v | ⊢ 𝑉 = (Base‘𝑈) |
hgmapvs.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hgmapvs.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hgmapvs.b | ⊢ 𝐵 = (Base‘𝑅) |
hgmapvs.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hgmapvs.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hgmapvs.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hgmapvs.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hgmapvs.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hgmapvs.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hgmapvs.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
hgmapvs | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hgmapvs.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | hgmapvs.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | hgmapvs.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | hgmapvs.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
5 | hgmapvs.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑈) | |
6 | hgmapvs.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
7 | hgmapvs.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
8 | hgmapvs.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
9 | hgmapvs.e | . . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
10 | hgmapvs.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
11 | hgmapvs.g | . . . . 5 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
12 | hgmapvs.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | hgmapvs.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
14 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | hgmapval 40350 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐹) = (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)))) |
15 | 14 | eqcomd 2742 | . . 3 ⊢ (𝜑 → (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) = (𝐺‘𝐹)) |
16 | 2, 3, 6, 7, 11, 12, 13 | hgmapcl 40352 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵) |
17 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13 | hdmap14lem15 40345 | . . . 4 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) |
18 | oveq1 7364 | . . . . . . 7 ⊢ (𝑔 = (𝐺‘𝐹) → (𝑔 ∙ (𝑆‘𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥))) | |
19 | 18 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑔 = (𝐺‘𝐹) → ((𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)) ↔ (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)))) |
20 | 19 | ralbidv 3174 | . . . . 5 ⊢ (𝑔 = (𝐺‘𝐹) → (∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)) ↔ ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)))) |
21 | 20 | riota2 7339 | . . . 4 ⊢ (((𝐺‘𝐹) ∈ 𝐵 ∧ ∃!𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) → (∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)) ↔ (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) = (𝐺‘𝐹))) |
22 | 16, 17, 21 | syl2anc 584 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)) ↔ (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) = (𝐺‘𝐹))) |
23 | 15, 22 | mpbird 256 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥))) |
24 | oveq2 7365 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 · 𝑥) = (𝐹 · 𝑋)) | |
25 | 24 | fveq2d 6846 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘(𝐹 · 𝑥)) = (𝑆‘(𝐹 · 𝑋))) |
26 | fveq2 6842 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋)) | |
27 | 26 | oveq2d 7373 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐺‘𝐹) ∙ (𝑆‘𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) |
28 | 25, 27 | eqeq12d 2752 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)) ↔ (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋)))) |
29 | 28 | rspcva 3579 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥))) → (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) |
30 | 1, 23, 29 | syl2anc 584 | 1 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃!wreu 3351 ‘cfv 6496 ℩crio 7312 (class class class)co 7357 Basecbs 17083 Scalarcsca 17136 ·𝑠 cvsca 17137 HLchlt 37812 LHypclh 38447 DVecHcdvh 39541 LCDualclcd 40049 HDMapchdma 40255 HGMapchg 40346 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lshyp 37439 df-lcv 37481 df-lfl 37520 df-lkr 37548 df-ldual 37586 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858 df-lcdual 40050 df-mapd 40088 df-hvmap 40220 df-hdmap1 40256 df-hdmap 40257 df-hgmap 40347 |
This theorem is referenced by: hgmapval0 40355 hgmapval1 40356 hgmapadd 40357 hgmapmul 40358 hgmaprnlem1N 40359 hgmap11 40365 hdmapglnm2 40374 |
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