![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh7cN | Structured version Visualization version GIF version |
Description: Part (7) of [Baer] p. 48 line 10 (3 of 6 cases). (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})) |
mapdh7a | ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑣〉) = 𝐺) |
Ref | Expression |
---|---|
mapdh7cN | ⊢ (𝜑 → (𝐼‘〈𝑣, 𝐺, 𝑢〉) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh7a | . 2 ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑣〉) = 𝐺) | |
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.y | . . 3 ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) | |
20 | 19 | eldifad 3921 | . . . . 5 ⊢ (𝜑 → 𝑣 ∈ 𝑉) |
21 | mapdh7.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) | |
22 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21 | mapdhcl 40179 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑣〉) ∈ 𝐷) |
23 | 1, 22 | eqeltrrd 2839 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
24 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 23, 21 | mapdheq2 40181 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑢, 𝐹, 𝑣〉) = 𝐺 → (𝐼‘〈𝑣, 𝐺, 𝑢〉) = 𝐹)) |
25 | 1, 24 | mpd 15 | 1 ⊢ (𝜑 → (𝐼‘〈𝑣, 𝐺, 𝑢〉) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2942 Vcvv 3444 ∖ cdif 3906 ifcif 4485 {csn 4585 {cpr 4587 〈cotp 4593 ↦ cmpt 5187 ‘cfv 6494 ℩crio 7309 (class class class)co 7354 1st c1st 7916 2nd c2nd 7917 Basecbs 17080 0gc0g 17318 -gcsg 18747 LSpanclspn 20428 HLchlt 37801 LHypclh 38436 DVecHcdvh 39530 LCDualclcd 40038 mapdcmpd 40076 |
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 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-riotaBAD 37404 |
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 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7614 df-om 7800 df-1st 7918 df-2nd 7919 df-tpos 8154 df-undef 8201 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-map 8764 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-n0 12411 df-z 12497 df-uz 12761 df-fz 13422 df-struct 17016 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-sca 17146 df-vsca 17147 df-0g 17320 df-mre 17463 df-mrc 17464 df-acs 17466 df-proset 18181 df-poset 18199 df-plt 18216 df-lub 18232 df-glb 18233 df-join 18234 df-meet 18235 df-p0 18311 df-p1 18312 df-lat 18318 df-clat 18385 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-submnd 18599 df-grp 18748 df-minusg 18749 df-sbg 18750 df-subg 18921 df-cntz 19093 df-oppg 19120 df-lsm 19414 df-cmn 19560 df-abl 19561 df-mgp 19893 df-ur 19910 df-ring 19962 df-oppr 20045 df-dvdsr 20066 df-unit 20067 df-invr 20097 df-dvr 20108 df-drng 20183 df-lmod 20320 df-lss 20389 df-lsp 20429 df-lvec 20560 df-lsatoms 37427 df-lshyp 37428 df-lcv 37470 df-lfl 37509 df-lkr 37537 df-ldual 37575 df-oposet 37627 df-ol 37629 df-oml 37630 df-covers 37717 df-ats 37718 df-atl 37749 df-cvlat 37773 df-hlat 37802 df-llines 37950 df-lplanes 37951 df-lvols 37952 df-lines 37953 df-psubsp 37955 df-pmap 37956 df-padd 38248 df-lhyp 38440 df-laut 38441 df-ldil 38556 df-ltrn 38557 df-trl 38611 df-tgrp 39195 df-tendo 39207 df-edring 39209 df-dveca 39455 df-disoa 39481 df-dvech 39531 df-dib 39591 df-dic 39625 df-dih 39681 df-doch 39800 df-djh 39847 df-lcdual 40039 df-mapd 40077 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |