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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem2N | Structured version Visualization version GIF version |
Description: Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 𝑍 so it can be used in hdmap14lem10 40737. (Contributed by NM, 31-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmap14lem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem1.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem1.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem3.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem1.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem1.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem1.z | ⊢ 𝑍 = (0g‘𝑅) |
hdmap14lem1.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem2.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem1.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap14lem2.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem2.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem2.q | ⊢ 𝑄 = (0g‘𝑃) |
hdmap14lem1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem3.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem1.f | ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) |
Ref | Expression |
---|---|
hdmap14lem2N | ⊢ (𝜑 → ∃𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem1.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap14lem1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap14lem1.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hdmap14lem3.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap14lem1.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
7 | hdmap14lem1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
8 | hdmap14lem1.z | . . . 4 ⊢ 𝑍 = (0g‘𝑅) | |
9 | hdmap14lem1.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
10 | hdmap14lem2.e | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
11 | hdmap14lem1.l | . . . 4 ⊢ 𝐿 = (LSpan‘𝐶) | |
12 | hdmap14lem2.p | . . . 4 ⊢ 𝑃 = (Scalar‘𝐶) | |
13 | hdmap14lem2.a | . . . 4 ⊢ 𝐴 = (Base‘𝑃) | |
14 | hdmap14lem2.q | . . . 4 ⊢ 𝑄 = (0g‘𝑃) | |
15 | hdmap14lem1.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
16 | hdmap14lem1.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hdmap14lem3.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
18 | hdmap14lem1.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | hdmap14lem1 40728 | . . 3 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) |
20 | 19 | eqcomd 2739 | . 2 ⊢ (𝜑 → (𝐿‘{(𝑆‘(𝐹 · 𝑋))}) = (𝐿‘{(𝑆‘𝑋)})) |
21 | eqid 2733 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
22 | 1, 9, 16 | lcdlvec 40451 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LVec) |
23 | 1, 2, 16 | dvhlmod 39970 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
24 | 18 | eldifad 3960 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
25 | 17 | eldifad 3960 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
26 | 3, 6, 4, 7 | lmodvscl 20482 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉) |
27 | 23, 24, 25, 26 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉) |
28 | 1, 2, 3, 9, 21, 15, 16, 27 | hdmapcl 40690 | . . 3 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) ∈ (Base‘𝐶)) |
29 | 1, 2, 3, 9, 21, 15, 16, 25 | hdmapcl 40690 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶)) |
30 | 21, 12, 13, 14, 10, 11, 22, 28, 29 | lspsneq 20728 | . 2 ⊢ (𝜑 → ((𝐿‘{(𝑆‘(𝐹 · 𝑋))}) = (𝐿‘{(𝑆‘𝑋)}) ↔ ∃𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))) |
31 | 20, 30 | mpbid 231 | 1 ⊢ (𝜑 → ∃𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ∖ cdif 3945 {csn 4628 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 Scalarcsca 17197 ·𝑠 cvsca 17198 0gc0g 17382 LModclmod 20464 LSpanclspn 20575 HLchlt 38209 LHypclh 38844 DVecHcdvh 39938 LCDualclcd 40446 HDMapchdma 40652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 37812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-0g 17384 df-mre 17527 df-mrc 17528 df-acs 17530 df-proset 18245 df-poset 18263 df-plt 18280 df-lub 18296 df-glb 18297 df-join 18298 df-meet 18299 df-p0 18375 df-p1 18376 df-lat 18382 df-clat 18449 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-subg 18998 df-cntz 19176 df-oppg 19205 df-lsm 19499 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-dvr 20208 df-drng 20310 df-lmod 20466 df-lss 20536 df-lsp 20576 df-lvec 20707 df-lsatoms 37835 df-lshyp 37836 df-lcv 37878 df-lfl 37917 df-lkr 37945 df-ldual 37983 df-oposet 38035 df-ol 38037 df-oml 38038 df-covers 38125 df-ats 38126 df-atl 38157 df-cvlat 38181 df-hlat 38210 df-llines 38358 df-lplanes 38359 df-lvols 38360 df-lines 38361 df-psubsp 38363 df-pmap 38364 df-padd 38656 df-lhyp 38848 df-laut 38849 df-ldil 38964 df-ltrn 38965 df-trl 39019 df-tgrp 39603 df-tendo 39615 df-edring 39617 df-dveca 39863 df-disoa 39889 df-dvech 39939 df-dib 39999 df-dic 40033 df-dih 40089 df-doch 40208 df-djh 40255 df-lcdual 40447 df-mapd 40485 df-hvmap 40617 df-hdmap1 40653 df-hdmap 40654 |
This theorem is referenced by: (None) |
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