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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 37092 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 3586 | . . 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 37088 | . 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 37092 | . . 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 37013 | . 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 2659 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐿‘〈𝑋, 𝐹, 𝑌〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ifcif 4086 {csn 4177 〈cotp 4185 ↦ cmpt 4729 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 Basecbs 15857 0gc0g 16100 -gcsg 17424 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913 HDMap1chdma1 37081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-1st 7168 df-2nd 7169 df-hdmap1 37083 |
This theorem is referenced by: hdmap1cl 37094 hdmap1eq2 37095 hdmap1eq4N 37096 hdmap1eulem 37113 hdmap1eulemOLDN 37114 |
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