![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 2738 | . . 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 3921 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18 | hdmap1val 40192 | . 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 4749 | . . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
21 | 20 | neneqd 2947 | . . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 ) |
22 | iffalse 4494 | . . 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 2778 | 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 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3906 ifcif 4485 {csn 4585 〈cotp 4593 ‘cfv 6492 ℩crio 7305 (class class class)co 7350 Basecbs 17019 0gc0g 17257 -gcsg 18686 LSpanclspn 20361 HLchlt 37743 LHypclh 38378 DVecHcdvh 39472 LCDualclcd 39980 mapdcmpd 40018 HDMap1chdma1 40185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-ot 4594 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-1st 7912 df-2nd 7913 df-hdmap1 40187 |
This theorem is referenced by: hdmap1eq 40195 |
Copyright terms: Public domain | W3C validator |