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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 6659 | . . . 4 ⊢ 0 ∈ V |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
8 | 1, 2, 3, 4, 7 | mapdhval 39020 | . 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 2798 | . . 3 ⊢ 0 = 0 | |
10 | 9 | iftruei 4432 | . 2 ⊢ if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)})))) = 𝑄 |
11 | 8, 10 | eqtrdi 2849 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ifcif 4425 {csn 4525 〈cotp 4533 ↦ cmpt 5110 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 0gc0g 16705 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-ov 7138 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: mapdhcl 39023 mapdh6bN 39033 mapdh6cN 39034 mapdh6dN 39035 mapdh8 39084 |
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