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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 6856 | . . . 4 ⊢ 0 ∈ V |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
8 | 1, 2, 3, 4, 7 | mapdhval 40177 | . 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 2736 | . . 3 ⊢ 0 = 0 | |
10 | 9 | iftruei 4493 | . 2 ⊢ if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)})))) = 𝑄 |
11 | 8, 10 | eqtrdi 2792 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ifcif 4486 {csn 4586 〈cotp 4594 ↦ cmpt 5188 ‘cfv 6496 ℩crio 7311 (class class class)co 7356 1st c1st 7918 2nd c2nd 7919 0gc0g 17320 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-ot 4595 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-iota 6448 df-fun 6498 df-fv 6504 df-riota 7312 df-ov 7359 df-1st 7920 df-2nd 7921 |
This theorem is referenced by: mapdhcl 40180 mapdh6bN 40190 mapdh6cN 40191 mapdh6dN 40192 mapdh8 40241 |
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