mapdhval0 - Mathbox for Norm Megill

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Theorem mapdhval0 39666

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 6770	. . . 4 ⊢ 0 ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
8	1, 2, 3, 4, 7	mapdhval 39665	. 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 2738	. . 3 ⊢ 0 = 0
10	9	iftruei 4463	. 2 ⊢ if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)})))) = 𝑄
11	8, 10	eqtrdi 2795	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ifcif 4456 {csn 4558 ⟨cotp 4566 ↦ cmpt 5153 ‘cfv 6418 ℩crio 7211 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 0_gc0g 17067

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-ot 4567 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fv 6426 df-riota 7212 df-ov 7258 df-1st 7804 df-2nd 7805

This theorem is referenced by: mapdhcl 39668 mapdh6bN 39678 mapdh6cN 39679 mapdh6dN 39680 mapdh8 39729

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