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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 37681 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐹) = (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)))) |
15 | 14 | eqcomd 2766 | . . 3 ⊢ (𝜑 → (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) = (𝐺‘𝐹)) |
16 | 2, 3, 6, 7, 11, 12, 13 | hgmapcl 37683 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵) |
17 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13 | hdmap14lem15 37676 | . . . 4 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) |
18 | oveq1 6820 | . . . . . . 7 ⊢ (𝑔 = (𝐺‘𝐹) → (𝑔 ∙ (𝑆‘𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥))) | |
19 | 18 | eqeq2d 2770 | . . . . . 6 ⊢ (𝑔 = (𝐺‘𝐹) → ((𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)) ↔ (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)))) |
20 | 19 | ralbidv 3124 | . . . . 5 ⊢ (𝑔 = (𝐺‘𝐹) → (∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)) ↔ ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)))) |
21 | 20 | riota2 6796 | . . . 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 696 | . . 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 247 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥))) |
24 | oveq2 6821 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 · 𝑥) = (𝐹 · 𝑋)) | |
25 | 24 | fveq2d 6356 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘(𝐹 · 𝑥)) = (𝑆‘(𝐹 · 𝑋))) |
26 | fveq2 6352 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋)) | |
27 | 26 | oveq2d 6829 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐺‘𝐹) ∙ (𝑆‘𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) |
28 | 25, 27 | eqeq12d 2775 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)) ↔ (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋)))) |
29 | 28 | rspcva 3447 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥))) → (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) |
30 | 1, 23, 29 | syl2anc 696 | 1 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ∃!wreu 3052 ‘cfv 6049 ℩crio 6773 (class class class)co 6813 Basecbs 16059 Scalarcsca 16146 ·𝑠 cvsca 16147 HLchlt 35140 LHypclh 35773 DVecHcdvh 36869 LCDualclcd 37377 HDMapchdma 37584 HGMapchg 37677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-riotaBAD 34742 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-tpos 7521 df-undef 7568 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-0g 16304 df-mre 16448 df-mrc 16449 df-acs 16451 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-p1 17241 df-lat 17247 df-clat 17309 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-subg 17792 df-cntz 17950 df-oppg 17976 df-lsm 18251 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-ring 18749 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-invr 18872 df-dvr 18883 df-drng 18951 df-lmod 19067 df-lss 19135 df-lsp 19174 df-lvec 19305 df-lsatoms 34766 df-lshyp 34767 df-lcv 34809 df-lfl 34848 df-lkr 34876 df-ldual 34914 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-llines 35287 df-lplanes 35288 df-lvols 35289 df-lines 35290 df-psubsp 35292 df-pmap 35293 df-padd 35585 df-lhyp 35777 df-laut 35778 df-ldil 35893 df-ltrn 35894 df-trl 35949 df-tgrp 36533 df-tendo 36545 df-edring 36547 df-dveca 36793 df-disoa 36820 df-dvech 36870 df-dib 36930 df-dic 36964 df-dih 37020 df-doch 37139 df-djh 37186 df-lcdual 37378 df-mapd 37416 df-hvmap 37548 df-hdmap1 37585 df-hdmap 37586 df-hgmap 37678 |
This theorem is referenced by: hgmapval0 37686 hgmapval1 37687 hgmapadd 37688 hgmapmul 37689 hgmaprnlem1N 37690 hgmap11 37696 hdmapglnm2 37705 |
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