mapdhval0 - Mathbox for Norm Megill

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Theorem mapdhval0 42598

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 6892	. . . 4 ⊢ 0 ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
8	1, 2, 3, 4, 7	mapdhval 42597	. 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 2760	. . 3 ⊢ 0 = 0
10	9	iftruei 4489	. 2 ⊢ if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)})))) = 𝑄
11	8, 10	eqtrdi 2811	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ifcif 4482 {csn 4584 ⟨cotp 4592 ↦ cmpt 5186 ‘cfv 6533 ℩crio 7369 (class class class)co 7413 1^st c1st 7984 2^nd c2nd 7985 0_gc0g 17524

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7736

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6489 df-fun 6535 df-fv 6541 df-riota 7370 df-ov 7416 df-1st 7986 df-2nd 7987

This theorem is used by: mapdhcl 42600 mapdh6bN 42610 mapdh6cN 42611 mapdh6dN 42612 mapdh8 42661

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