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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 2737 | . . 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 3963 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| 19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18 | hdmap1val 41800 | . 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 4790 | . . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
| 21 | 20 | neneqd 2945 | . . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 ) |
| 22 | iffalse 4534 | . . 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 2777 | 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 1540 ∈ wcel 2108 ∖ cdif 3948 ifcif 4525 {csn 4626 〈cotp 4634 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 0gc0g 17484 -gcsg 18953 LSpanclspn 20969 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 LCDualclcd 41588 mapdcmpd 41626 HDMap1chdma1 41793 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-1st 8014 df-2nd 8015 df-hdmap1 41795 |
| This theorem is referenced by: hdmap1eq 41803 |
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