Theorem mapdhval2 42385

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

Hypotheses

Ref	Expression
mapdh.q	⊢ 𝑄 = (0g‘𝐶)
mapdh.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))
mapdh2.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
mapdh2.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
mapdh2.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))

Assertion

Ref	Expression
mapdhval2	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))

Distinct variable groups:   𝑥,𝐷   𝑥,ℎ,𝐹   𝑥,𝐽   𝑥,𝑀   𝑥,𝑁   𝑥, 0   𝑥,𝑄   𝑥,𝑅   𝑥, −   ℎ,𝑋,𝑥   ℎ,𝑌,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)   𝐼(𝑥,ℎ)   𝐽(ℎ)   𝑀(ℎ)   − (ℎ)   𝑁(ℎ)   𝑉(𝑥,ℎ)

Proof of Theorem mapdhval2

Step	Hyp	Ref	Expression
1		mapdh.q	. . 3 ⊢ 𝑄 = (0g‘𝐶)
2		mapdh.i	. . 3 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))
3		mapdh2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4		mapdh2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
5		mapdh2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
6	1, 2, 3, 4, 5	mapdhval 42383	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))
7		eldifsni 4759	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
8	7	neneqd 2969	. . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
9		iffalse 4498	. . 3 ⊢ (¬ 𝑌 = 0 → if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
10	5, 8, 9	3syl 19	. 2 ⊢ (𝜑 → if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
11	6, 10	eqtrd 2804	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910  ifcif 4489  {csn 4591  ⟨cotp 4599   ↦ cmpt 5193  ‘cfv 6533  ℩crio 7364  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  0_gc0g 17488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6535  df-fv 6541  df-riota 7365  df-ov 7411  df-1st 7982  df-2nd 7983

This theorem is referenced by:  mapdhcl  42386  mapdheq  42387