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Mathbox for Norm Megill |
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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 2735 | . . 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 3975 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18 | hdmap1val 41781 | . 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 4795 | . . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
21 | 20 | neneqd 2943 | . . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 ) |
22 | iffalse 4540 | . . 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 2775 | 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 395 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ifcif 4531 {csn 4631 〈cotp 4639 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 Basecbs 17245 0gc0g 17486 -gcsg 18966 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607 HDMap1chdma1 41774 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-1st 8013 df-2nd 8014 df-hdmap1 41776 |
This theorem is referenced by: hdmap1eq 41784 |
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