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 40113 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐹) = (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)))) |
15 | 14 | eqcomd 2743 | . . 3 ⊢ (𝜑 → (℩𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) = (𝐺‘𝐹)) |
16 | 2, 3, 6, 7, 11, 12, 13 | hgmapcl 40115 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵) |
17 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13 | hdmap14lem15 40108 | . . . 4 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) |
18 | oveq1 7320 | . . . . . . 7 ⊢ (𝑔 = (𝐺‘𝐹) → (𝑔 ∙ (𝑆‘𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥))) | |
19 | 18 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑔 = (𝐺‘𝐹) → ((𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)) ↔ (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)))) |
20 | 19 | ralbidv 3171 | . . . . 5 ⊢ (𝑔 = (𝐺‘𝐹) → (∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)) ↔ ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)))) |
21 | 20 | riota2 7296 | . . . 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 7321 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 · 𝑥) = (𝐹 · 𝑋)) | |
25 | 24 | fveq2d 6813 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘(𝐹 · 𝑥)) = (𝑆‘(𝐹 · 𝑋))) |
26 | fveq2 6809 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋)) | |
27 | 26 | oveq2d 7329 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐺‘𝐹) ∙ (𝑆‘𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) |
28 | 25, 27 | eqeq12d 2753 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑆‘(𝐹 · 𝑥)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑥)) ↔ (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋)))) |
29 | 28 | rspcva 3568 | . 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 1540 ∈ wcel 2105 ∀wral 3062 ∃!wreu 3348 ‘cfv 6463 ℩crio 7269 (class class class)co 7313 Basecbs 16979 Scalarcsca 17032 ·𝑠 cvsca 17033 HLchlt 37576 LHypclh 38210 DVecHcdvh 39304 LCDualclcd 39812 HDMapchdma 40018 HGMapchg 40109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5222 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018 ax-riotaBAD 37179 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4849 df-int 4891 df-iun 4937 df-iin 4938 df-br 5086 df-opab 5148 df-mpt 5169 df-tr 5203 df-id 5505 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5560 df-we 5562 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-pred 6222 df-ord 6289 df-on 6290 df-lim 6291 df-suc 6292 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-of 7571 df-om 7756 df-1st 7874 df-2nd 7875 df-tpos 8087 df-undef 8134 df-frecs 8142 df-wrecs 8173 df-recs 8247 df-rdg 8286 df-1o 8342 df-er 8544 df-map 8663 df-en 8780 df-dom 8781 df-sdom 8782 df-fin 8783 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-nn 12044 df-2 12106 df-3 12107 df-4 12108 df-5 12109 df-6 12110 df-n0 12304 df-z 12390 df-uz 12653 df-fz 13310 df-struct 16915 df-sets 16932 df-slot 16950 df-ndx 16962 df-base 16980 df-ress 17009 df-plusg 17042 df-mulr 17043 df-sca 17045 df-vsca 17046 df-0g 17219 df-mre 17362 df-mrc 17363 df-acs 17365 df-proset 18080 df-poset 18098 df-plt 18115 df-lub 18131 df-glb 18132 df-join 18133 df-meet 18134 df-p0 18210 df-p1 18211 df-lat 18217 df-clat 18284 df-mgm 18393 df-sgrp 18442 df-mnd 18453 df-submnd 18498 df-grp 18647 df-minusg 18648 df-sbg 18649 df-subg 18819 df-cntz 18990 df-oppg 19017 df-lsm 19308 df-cmn 19455 df-abl 19456 df-mgp 19788 df-ur 19805 df-ring 19852 df-oppr 19929 df-dvdsr 19950 df-unit 19951 df-invr 19981 df-dvr 19992 df-drng 20064 df-lmod 20196 df-lss 20265 df-lsp 20305 df-lvec 20436 df-lsatoms 37202 df-lshyp 37203 df-lcv 37245 df-lfl 37284 df-lkr 37312 df-ldual 37350 df-oposet 37402 df-ol 37404 df-oml 37405 df-covers 37492 df-ats 37493 df-atl 37524 df-cvlat 37548 df-hlat 37577 df-llines 37724 df-lplanes 37725 df-lvols 37726 df-lines 37727 df-psubsp 37729 df-pmap 37730 df-padd 38022 df-lhyp 38214 df-laut 38215 df-ldil 38330 df-ltrn 38331 df-trl 38385 df-tgrp 38969 df-tendo 38981 df-edring 38983 df-dveca 39229 df-disoa 39255 df-dvech 39305 df-dib 39365 df-dic 39399 df-dih 39455 df-doch 39574 df-djh 39621 df-lcdual 39813 df-mapd 39851 df-hvmap 39983 df-hdmap1 40019 df-hdmap 40020 df-hgmap 40110 |
This theorem is referenced by: hgmapval0 40118 hgmapval1 40119 hgmapadd 40120 hgmapmul 40121 hgmaprnlem1N 40122 hgmap11 40128 hdmapglnm2 40137 |
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