Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mapdhval Structured version   Visualization version   GIF version

Theorem mapdhval 37534
Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-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)})))))
mapdh.x (𝜑𝑋𝐴)
mapdh.f (𝜑𝐹𝐵)
mapdh.y (𝜑𝑌𝐸)
Assertion
Ref Expression
mapdhval (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (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,
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,)   𝐵(𝑥,)   𝐶(𝑥,)   𝐷()   𝑄()   𝑅()   𝐸(𝑥,)   𝐼(𝑥,)   𝐽()   𝑀()   ()   𝑁()   0 ()

Proof of Theorem mapdhval
StepHypRef Expression
1 otex 5061 . . 3 𝑋, 𝐹, 𝑌⟩ ∈ V
2 fveq2 6332 . . . . . 6 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (2nd𝑥) = (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))
32eqeq1d 2773 . . . . 5 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((2nd𝑥) = 0 ↔ (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ))
42sneqd 4328 . . . . . . . . . 10 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {(2nd𝑥)} = {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})
54fveq2d 6336 . . . . . . . . 9 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑁‘{(2nd𝑥)}) = (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}))
65fveq2d 6336 . . . . . . . 8 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑀‘(𝑁‘{(2nd𝑥)})) = (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})))
76eqeq1d 2773 . . . . . . 7 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{})))
8 fveq2 6332 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (1st𝑥) = (1st ‘⟨𝑋, 𝐹, 𝑌⟩))
98fveq2d 6336 . . . . . . . . . . . 12 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (1st ‘(1st𝑥)) = (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)))
109, 2oveq12d 6811 . . . . . . . . . . 11 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((1st ‘(1st𝑥)) (2nd𝑥)) = ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)))
1110sneqd 4328 . . . . . . . . . 10 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {((1st ‘(1st𝑥)) (2nd𝑥))} = {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})
1211fveq2d 6336 . . . . . . . . 9 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))}) = (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}))
1312fveq2d 6336 . . . . . . . 8 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})))
148fveq2d 6336 . . . . . . . . . . 11 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (2nd ‘(1st𝑥)) = (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)))
1514oveq1d 6808 . . . . . . . . . 10 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((2nd ‘(1st𝑥))𝑅) = ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅))
1615sneqd 4328 . . . . . . . . 9 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {((2nd ‘(1st𝑥))𝑅)} = {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})
1716fveq2d 6336 . . . . . . . 8 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))
1813, 17eqeq12d 2786 . . . . . . 7 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) ↔ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))
197, 18anbi12d 608 . . . . . 6 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})) ↔ ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))))
2019riotabidv 6756 . . . . 5 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))) = (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))))
213, 20ifbieq2d 4250 . . . 4 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))))
22 mapdh.i . . . 4 𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
23 mapdh.q . . . . . 6 𝑄 = (0g𝐶)
24 fvex 6342 . . . . . 6 (0g𝐶) ∈ V
2523, 24eqeltri 2846 . . . . 5 𝑄 ∈ V
26 riotaex 6758 . . . . 5 (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))) ∈ V
2725, 26ifex 4295 . . . 4 if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))) ∈ V
2821, 22, 27fvmpt 6424 . . 3 (⟨𝑋, 𝐹, 𝑌⟩ ∈ V → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))))
291, 28mp1i 13 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))))
30 mapdh.y . . . . 5 (𝜑𝑌𝐸)
31 ot3rdg 7331 . . . . 5 (𝑌𝐸 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
3230, 31syl 17 . . . 4 (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
3332eqeq1d 2773 . . 3 (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0𝑌 = 0 ))
3432sneqd 4328 . . . . . . . 8 (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
3534fveq2d 6336 . . . . . . 7 (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
3635fveq2d 6336 . . . . . 6 (𝜑 → (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝑀‘(𝑁‘{𝑌})))
3736eqeq1d 2773 . . . . 5 (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{})))
38 mapdh.x . . . . . . . . . . 11 (𝜑𝑋𝐴)
39 mapdh.f . . . . . . . . . . 11 (𝜑𝐹𝐵)
40 ot1stg 7329 . . . . . . . . . . 11 ((𝑋𝐴𝐹𝐵𝑌𝐸) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
4138, 39, 30, 40syl3anc 1476 . . . . . . . . . 10 (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
4241, 32oveq12d 6811 . . . . . . . . 9 (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 𝑌))
4342sneqd 4328 . . . . . . . 8 (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 𝑌)})
4443fveq2d 6336 . . . . . . 7 (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 𝑌)}))
4544fveq2d 6336 . . . . . 6 (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 𝑌)})))
46 ot2ndg 7330 . . . . . . . . . 10 ((𝑋𝐴𝐹𝐵𝑌𝐸) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
4738, 39, 30, 46syl3anc 1476 . . . . . . . . 9 (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
4847oveq1d 6808 . . . . . . . 8 (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅) = (𝐹𝑅))
4948sneqd 4328 . . . . . . 7 (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)} = {(𝐹𝑅)})
5049fveq2d 6336 . . . . . 6 (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}) = (𝐽‘{(𝐹𝑅)}))
5145, 50eqeq12d 2786 . . . . 5 (𝜑 → ((𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}) ↔ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))
5237, 51anbi12d 608 . . . 4 (𝜑 → (((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
5352riotabidv 6756 . . 3 (𝜑 → (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
5433, 53ifbieq2d 4250 . 2 (𝜑 → if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
5529, 54eqtrd 2805 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  ifcif 4225  {csn 4316  cotp 4324  cmpt 4863  cfv 6031  crio 6753  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  0gc0g 16308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-ot 4325  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-riota 6754  df-ov 6796  df-1st 7315  df-2nd 7316
This theorem is referenced by:  mapdhval0  37535  mapdhval2  37536  hdmap1valc  37613
  Copyright terms: Public domain W3C validator