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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1valc | Structured version Visualization version GIF version |
Description: Connect the value of the preliminary map from vectors to functionals 𝐼 to the hypothesis 𝐿 used by earlier theorems. Note: the 𝑋 ∈ (𝑉 ∖ { 0 }) hypothesis could be the more general 𝑋 ∈ 𝑉 but the former will be easier to use. TODO: use the 𝐼 function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 39098 is probably unnecessary, but it would mean different $d's later on. (Contributed by NM, 15-May-2015.) |
Ref | Expression |
---|---|
hdmap1valc.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap1valc.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap1valc.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap1valc.s | ⊢ − = (-g‘𝑈) |
hdmap1valc.o | ⊢ 0 = (0g‘𝑈) |
hdmap1valc.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap1valc.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap1valc.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap1valc.r | ⊢ 𝑅 = (-g‘𝐶) |
hdmap1valc.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmap1valc.j | ⊢ 𝐽 = (LSpan‘𝐶) |
hdmap1valc.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap1valc.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap1valc.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap1valc.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap1valc.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
hdmap1valc.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
hdmap1valc.l | ⊢ 𝐿 = (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)}))))) |
Ref | Expression |
---|---|
hdmap1valc | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐿‘〈𝑋, 𝐹, 𝑌〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1valc.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1valc.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1valc.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap1valc.s | . . 3 ⊢ − = (-g‘𝑈) | |
5 | hdmap1valc.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap1valc.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | hdmap1valc.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap1valc.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
9 | hdmap1valc.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
10 | hdmap1valc.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
11 | hdmap1valc.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | hdmap1valc.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | hdmap1valc.i | . . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
14 | hdmap1valc.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | hdmap1valc.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
16 | 15 | eldifad 3893 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
17 | hdmap1valc.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
18 | hdmap1valc.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18 | hdmap1val 39094 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = if(𝑌 = 0 , 𝑄, (℩𝑔 ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝑔)}))))) |
20 | hdmap1valc.l | . . . 4 ⊢ 𝐿 = (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)}))))) | |
21 | 20 | hdmap1cbv 39098 | . . 3 ⊢ 𝐿 = (𝑤 ∈ V ↦ if((2nd ‘𝑤) = 0 , 𝑄, (℩𝑔 ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑤)})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑤)) − (2nd ‘𝑤))})) = (𝐽‘{((2nd ‘(1st ‘𝑤))𝑅𝑔)}))))) |
22 | 10, 21, 16, 17, 18 | mapdhval 39020 | . 2 ⊢ (𝜑 → (𝐿‘〈𝑋, 𝐹, 𝑌〉) = if(𝑌 = 0 , 𝑄, (℩𝑔 ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝑔)}))))) |
23 | 19, 22 | eqtr4d 2836 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐿‘〈𝑋, 𝐹, 𝑌〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ifcif 4425 {csn 4525 〈cotp 4533 ↦ cmpt 5110 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 Basecbs 16475 0gc0g 16705 -gcsg 18097 LSpanclspn 19736 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 LCDualclcd 38882 mapdcmpd 38920 HDMap1chdma1 39087 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-1st 7671 df-2nd 7672 df-hdmap1 39089 |
This theorem is referenced by: hdmap1cl 39100 hdmap1eq2 39101 hdmap1eq4N 39102 hdmap1eulem 39118 hdmap1eulemOLDN 39119 |
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