mapdhval0 - Mathbox for Norm Megill

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Theorem mapdhval0 38876

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 6684	. . . 4 ⊢ 0 ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
8	1, 2, 3, 4, 7	mapdhval 38875	. 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 2821	. . 3 ⊢ 0 = 0
10	9	iftruei 4474	. 2 ⊢ if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)})))) = 𝑄
11	8, 10	syl6eq 2872	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ifcif 4467 {csn 4567 ⟨cotp 4575 ↦ cmpt 5146 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 1^st c1st 7687 2^nd c2nd 7688 0_gc0g 16713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-ov 7159 df-1st 7689 df-2nd 7690

This theorem is referenced by: mapdhcl 38878 mapdh6bN 38888 mapdh6cN 38889 mapdh6dN 38890 mapdh8 38939

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