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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 6460 | . . . 4 ⊢ 0 ∈ V |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
8 | 1, 2, 3, 4, 7 | mapdhval 37878 | . 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 2778 | . . 3 ⊢ 0 = 0 | |
10 | 9 | iftruei 4314 | . 2 ⊢ if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)})))) = 𝑄 |
11 | 8, 10 | syl6eq 2830 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ifcif 4307 {csn 4398 〈cotp 4406 ↦ cmpt 4965 ‘cfv 6135 ℩crio 6882 (class class class)co 6922 1st c1st 7443 2nd c2nd 7444 0gc0g 16486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-ot 4407 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fv 6143 df-riota 6883 df-ov 6925 df-1st 7445 df-2nd 7446 |
This theorem is referenced by: mapdhcl 37881 mapdh6bN 37891 mapdh6cN 37892 mapdh6dN 37893 mapdh8 37942 |
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