mapdhval0 - Mathbox for Norm Megill

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Theorem mapdhval0 42480

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 6897	. . . 4 ⊢ 0 ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
8	1, 2, 3, 4, 7	mapdhval 42479	. 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 2763	. . 3 ⊢ 0 = 0
10	9	iftruei 4495	. 2 ⊢ if( 0 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 0 )})) = (𝐽‘{(𝐹𝑅ℎ)})))) = 𝑄
11	8, 10	eqtrdi 2814	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ifcif 4488 {csn 4590 ⟨cotp 4598 ↦ cmpt 5193 ‘cfv 6538 ℩crio 7368 (class class class)co 7412 1^st c1st 7985 2^nd c2nd 7986 0_gc0g 17493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-iota 6494 df-fun 6540 df-fv 6546 df-riota 7369 df-ov 7415 df-1st 7987 df-2nd 7988

This theorem is referenced by: mapdhcl 42482 mapdh6bN 42492 mapdh6cN 42493 mapdh6dN 42494 mapdh8 42543

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