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Theorem mapdhval2 41427
Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
Hypotheses
Ref Expression
mapdh.q 𝑄 = (0g𝐶)
mapdh.i 𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
mapdh2.x (𝜑𝑋𝐴)
mapdh2.f (𝜑𝐹𝐵)
mapdh2.y (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
Assertion
Ref Expression
mapdhval2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
Distinct variable groups:   𝑥,𝐷   𝑥,,𝐹   𝑥,𝐽   𝑥,𝑀   𝑥,𝑁   𝑥, 0   𝑥,𝑄   𝑥,𝑅   𝑥,   ,𝑋,𝑥   ,𝑌,𝑥   𝜑,   0 ,
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,)   𝐵(𝑥,)   𝐶(𝑥,)   𝐷()   𝑄()   𝑅()   𝐼(𝑥,)   𝐽()   𝑀()   ()   𝑁()   𝑉(𝑥,)

Proof of Theorem mapdhval2
StepHypRef Expression
1 mapdh.q . . 3 𝑄 = (0g𝐶)
2 mapdh.i . . 3 𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
3 mapdh2.x . . 3 (𝜑𝑋𝐴)
4 mapdh2.f . . 3 (𝜑𝐹𝐵)
5 mapdh2.y . . 3 (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
61, 2, 3, 4, 5mapdhval 41425 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
7 eldifsni 4799 . . . 4 (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌0 )
87neneqd 2935 . . 3 (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
9 iffalse 4542 . . 3 𝑌 = 0 → if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
105, 8, 93syl 18 . 2 (𝜑 → if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
116, 10eqtrd 2766 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  cdif 3944  ifcif 4533  {csn 4633  cotp 4641  cmpt 5238  cfv 6556  crio 7381  (class class class)co 7426  1st c1st 8003  2nd c2nd 8004  0gc0g 17456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-ot 4642  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6508  df-fun 6558  df-fv 6564  df-riota 7382  df-ov 7429  df-1st 8005  df-2nd 8006
This theorem is referenced by:  mapdhcl  41428  mapdheq  41429
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