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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh7eN | Structured version Visualization version GIF version |
Description: Part (7) of [Baer] p. 48 line 10 (5 of 6 cases). (Note: 1 of 6 and 2 of 6 are hypotheses a and b.) (Contributed by NM, 2-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdh7.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh7.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh7.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh7.s | ⊢ − = (-g‘𝑈) |
mapdh7.o | ⊢ 0 = (0g‘𝑈) |
mapdh7.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh7.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh7.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh7.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh7.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh7.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh7.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh7.i | ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) |
mapdh7.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdh7.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh7.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) |
mapdh7.x | ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) |
mapdh7.y | ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) |
mapdh7.z | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdh7.ne | ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) |
mapdh7.wn | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣})) |
mapdh7b | ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑤〉) = 𝐸) |
Ref | Expression |
---|---|
mapdh7eN | ⊢ (𝜑 → (𝐼‘〈𝑤, 𝐸, 𝑢〉) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh7b | . 2 ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑤〉) = 𝐸) | |
2 | mapdh7.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
3 | mapdh7.i | . . 3 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) | |
4 | mapdh7.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | mapdh7.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
6 | mapdh7.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | mapdh7.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
8 | mapdh7.s | . . 3 ⊢ − = (-g‘𝑈) | |
9 | mapdh7.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
10 | mapdh7.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
11 | mapdh7.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
12 | mapdh7.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
13 | mapdh7.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
14 | mapdh7.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
15 | mapdh7.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | mapdh7.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
17 | mapdh7.mn | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) | |
18 | mapdh7.x | . . 3 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) | |
19 | mapdh7.z | . . 3 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
20 | 19 | eldifad 3959 | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
21 | 4, 6, 15 | dvhlvec 39918 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | 18 | eldifad 3959 | . . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ 𝑉) |
23 | mapdh7.y | . . . . . . . . 9 ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) | |
24 | 23 | eldifad 3959 | . . . . . . . 8 ⊢ (𝜑 → 𝑣 ∈ 𝑉) |
25 | mapdh7.wn | . . . . . . . 8 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣})) | |
26 | 7, 10, 21, 20, 22, 24, 25 | lspindpi 20733 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣}))) |
27 | 26 | simpld 496 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑢})) |
28 | 27 | necomd 2997 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑤})) |
29 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 28 | mapdhcl 40536 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑤〉) ∈ 𝐷) |
30 | 1, 29 | eqeltrrd 2835 | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐷) |
31 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 28 | mapdheq2 40538 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑢, 𝐹, 𝑤〉) = 𝐸 → (𝐼‘〈𝑤, 𝐸, 𝑢〉) = 𝐹)) |
32 | 1, 31 | mpd 15 | 1 ⊢ (𝜑 → (𝐼‘〈𝑤, 𝐸, 𝑢〉) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∖ cdif 3944 ifcif 4527 {csn 4627 {cpr 4629 〈cotp 4635 ↦ cmpt 5230 ‘cfv 6540 ℩crio 7359 (class class class)co 7404 1st c1st 7968 2nd c2nd 7969 Basecbs 17140 0gc0g 17381 -gcsg 18817 LSpanclspn 20570 HLchlt 38158 LHypclh 38793 DVecHcdvh 39887 LCDualclcd 40395 mapdcmpd 40433 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37761 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-mre 17526 df-mrc 17527 df-acs 17529 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-oppg 19203 df-lsm 19497 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-drng 20306 df-lmod 20461 df-lss 20531 df-lsp 20571 df-lvec 20702 df-lsatoms 37784 df-lshyp 37785 df-lcv 37827 df-lfl 37866 df-lkr 37894 df-ldual 37932 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-atl 38106 df-cvlat 38130 df-hlat 38159 df-llines 38307 df-lplanes 38308 df-lvols 38309 df-lines 38310 df-psubsp 38312 df-pmap 38313 df-padd 38605 df-lhyp 38797 df-laut 38798 df-ldil 38913 df-ltrn 38914 df-trl 38968 df-tgrp 39552 df-tendo 39564 df-edring 39566 df-dveca 39812 df-disoa 39838 df-dvech 39888 df-dib 39948 df-dic 39982 df-dih 40038 df-doch 40157 df-djh 40204 df-lcdual 40396 df-mapd 40434 |
This theorem is referenced by: (None) |
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