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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 37409 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 3619 | . . 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 37405 | . 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 37409 | . . 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 37330 | . 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 2688 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐿‘〈𝑋, 𝐹, 𝑌〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∖ cdif 3604 ifcif 4119 {csn 4210 〈cotp 4218 ↦ cmpt 4762 ‘cfv 5926 ℩crio 6650 (class class class)co 6690 1st c1st 7208 2nd c2nd 7209 Basecbs 15904 0gc0g 16147 -gcsg 17471 LSpanclspn 19019 HLchlt 34955 LHypclh 35588 DVecHcdvh 36684 LCDualclcd 37192 mapdcmpd 37230 HDMap1chdma1 37398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-ot 4219 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-1st 7210 df-2nd 7211 df-hdmap1 37400 |
This theorem is referenced by: hdmap1cl 37411 hdmap1eq2 37412 hdmap1eq4N 37413 hdmap1eulem 37430 hdmap1eulemOLDN 37431 |
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