mapdhval0 - Mathbox for Norm Megill

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Theorem mapdhval0 37879

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)

**Hypotheses**
Ref	Expression
mapdh.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
mapdh0.o	⊢ 0 = (0_g‘𝑈)
mapdh0.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
mapdh0.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)

**Assertion**
Ref	Expression
mapdhval0	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Distinct variable groups: 𝑥,𝐷 𝑥,ℎ,𝐹 𝑥,𝐽 𝑥,𝑀 𝑥,𝑁 𝑥, 0 𝑥,𝑄 𝑥,𝑅 𝑥, − ℎ,𝑋,𝑥 𝜑,ℎ 0 ,ℎ Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑥,ℎ) 𝐵(𝑥,ℎ) 𝐶(𝑥,ℎ) 𝐷(ℎ) 𝑄(ℎ) 𝑅(ℎ) 𝑈(𝑥,ℎ) 𝐼(𝑥,ℎ) 𝐽(ℎ) 𝑀(ℎ) − (ℎ) 𝑁(ℎ)

**Proof of Theorem mapdhval0**
Step	Hyp	Ref	Expression
1		mapdh.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
2		mapdh.i	. . 3 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
3		mapdh0.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4		mapdh0.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
5		mapdh0.o	. . . . 5 ⊢ 0 = (0_g‘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 1^st c1st 7443 2^nd c2nd 7444 0_gc0g 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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