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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdhval0 | Structured version Visualization version GIF version |
Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
mapdh.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh.i | ⊢ 𝐼 = (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)}))))) |
mapdh0.o | ⊢ 0 = (0g‘𝑈) |
mapdh0.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
mapdh0.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
mapdhval0 | ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
2 | mapdh.i | . . 3 ⊢ 𝐼 = (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)}))))) | |
3 | mapdh0.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
4 | mapdh0.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
5 | mapdh0.o | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
6 | 5 | fvexi 6898 | . . . 4 ⊢ 0 ∈ V |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
8 | 1, 2, 3, 4, 7 | mapdhval 41106 | . 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)}))))) |
9 | eqid 2726 | . . 3 ⊢ 0 = 0 | |
10 | 9 | iftruei 4530 | . 2 ⊢ if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)})))) = 𝑄 |
11 | 8, 10 | eqtrdi 2782 | 1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ifcif 4523 {csn 4623 ⟨cotp 4631 ↦ cmpt 5224 ‘cfv 6536 ℩crio 7359 (class class class)co 7404 1st c1st 7969 2nd c2nd 7970 0gc0g 17392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-ot 4632 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fv 6544 df-riota 7360 df-ov 7407 df-1st 7971 df-2nd 7972 |
This theorem is referenced by: mapdhcl 41109 mapdh6bN 41119 mapdh6cN 41120 mapdh6dN 41121 mapdh8 41170 |
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