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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 38932 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 3948 | . . 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 38928 | . 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 38932 | . . 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 38854 | . 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 2859 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐿‘〈𝑋, 𝐹, 𝑌〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∖ cdif 3933 ifcif 4467 {csn 4561 〈cotp 4569 ↦ cmpt 5139 ‘cfv 6350 ℩crio 7107 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 Basecbs 16477 0gc0g 16707 -gcsg 18099 LSpanclspn 19737 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 LCDualclcd 38716 mapdcmpd 38754 HDMap1chdma1 38921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-ot 4570 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-1st 7683 df-2nd 7684 df-hdmap1 38923 |
This theorem is referenced by: hdmap1cl 38934 hdmap1eq2 38935 hdmap1eq4N 38936 hdmap1eulem 38952 hdmap1eulemOLDN 38953 |
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