Theorem mapdhval2 39940

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 39938	. 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 4729	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
8	7	neneqd 2946	. . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
9		iffalse 4474	. . 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 18	. 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 2776	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 397   = wceq 1539   ∈ wcel 2104  Vcvv 3437   ∖ cdif 3889  ifcif 4465  {csn 4565  ⟨cotp 4573   ↦ cmpt 5164  ‘cfv 6458  ℩crio 7263  (class class class)co 7307  1^st c1st 7861  2^nd c2nd 7862  0_gc0g 17199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-ot 4574  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-iota 6410  df-fun 6460  df-fv 6466  df-riota 7264  df-ov 7310  df-1st 7863  df-2nd 7864

This theorem is referenced by:  mapdhcl  39941  mapdheq  39942