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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1val2 | Structured version Visualization version GIF version |
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero 𝑌. (Contributed by NM, 16-May-2015.) |
Ref | Expression |
---|---|
hdmap1val2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap1val2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap1val2.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap1val2.s | ⊢ − = (-g‘𝑈) |
hdmap1val2.o | ⊢ 0 = (0g‘𝑈) |
hdmap1val2.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap1val2.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap1val2.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap1val2.r | ⊢ 𝑅 = (-g‘𝐶) |
hdmap1val2.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap1val2.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap1val2.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap1val2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap1val2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmap1val2.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
hdmap1val2.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
hdmap1val2 | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1val2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1val2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1val2.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap1val2.s | . . 3 ⊢ − = (-g‘𝑈) | |
5 | hdmap1val2.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap1val2.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | hdmap1val2.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap1val2.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
9 | hdmap1val2.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
10 | eqid 2798 | . . 3 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
11 | hdmap1val2.l | . . 3 ⊢ 𝐿 = (LSpan‘𝐶) | |
12 | hdmap1val2.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | hdmap1val2.i | . . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
14 | hdmap1val2.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | hdmap1val2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
16 | hdmap1val2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
17 | hdmap1val2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
18 | 17 | eldifad 3893 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18 | hdmap1val 39094 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = if(𝑌 = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)}))))) |
20 | eldifsni 4683 | . . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
21 | 20 | neneqd 2992 | . . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 ) |
22 | iffalse 4434 | . . 3 ⊢ (¬ 𝑌 = 0 → if(𝑌 = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) | |
23 | 17, 21, 22 | 3syl 18 | . 2 ⊢ (𝜑 → if(𝑌 = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) |
24 | 19, 23 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ifcif 4425 {csn 4525 〈cotp 4533 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 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: hdmap1eq 39097 |
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