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Theorem fvmpopr2d 7595
Description: Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.)
Hypotheses
Ref Expression
fvmpopr2d.1 (𝜑𝐹 = (𝑎𝐴, 𝑏𝐵𝐶))
fvmpopr2d.2 (𝜑𝑃 = ⟨𝑎, 𝑏⟩)
fvmpopr2d.3 ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
Assertion
Ref Expression
fvmpopr2d ((𝜑𝑎𝐴𝑏𝐵) → (𝐹𝑃) = 𝐶)
Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐶(𝑎,𝑏)   𝑃(𝑎,𝑏)   𝐹(𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem fvmpopr2d
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7434 . . 3 (𝑎(𝑎𝐴, 𝑏𝐵𝐶)𝑏) = ((𝑎𝐴, 𝑏𝐵𝐶)‘⟨𝑎, 𝑏⟩)
2 fvmpopr2d.1 . . . . 5 (𝜑𝐹 = (𝑎𝐴, 𝑏𝐵𝐶))
323ad2ant1 1134 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝐹 = (𝑎𝐴, 𝑏𝐵𝐶))
4 fvmpopr2d.2 . . . . 5 (𝜑𝑃 = ⟨𝑎, 𝑏⟩)
543ad2ant1 1134 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝑃 = ⟨𝑎, 𝑏⟩)
63, 5fveq12d 6913 . . 3 ((𝜑𝑎𝐴𝑏𝐵) → (𝐹𝑃) = ((𝑎𝐴, 𝑏𝐵𝐶)‘⟨𝑎, 𝑏⟩))
71, 6eqtr4id 2796 . 2 ((𝜑𝑎𝐴𝑏𝐵) → (𝑎(𝑎𝐴, 𝑏𝐵𝐶)𝑏) = (𝐹𝑃))
8 nfcv 2905 . . . . 5 𝑐𝐶
9 nfcv 2905 . . . . 5 𝑑𝐶
10 nfcv 2905 . . . . . 6 𝑎𝑑
11 nfcsb1v 3923 . . . . . 6 𝑎𝑐 / 𝑎𝐶
1210, 11nfcsbw 3925 . . . . 5 𝑎𝑑 / 𝑏𝑐 / 𝑎𝐶
13 nfcsb1v 3923 . . . . 5 𝑏𝑑 / 𝑏𝑐 / 𝑎𝐶
14 csbeq1a 3913 . . . . . 6 (𝑎 = 𝑐𝐶 = 𝑐 / 𝑎𝐶)
15 csbeq1a 3913 . . . . . 6 (𝑏 = 𝑑𝑐 / 𝑎𝐶 = 𝑑 / 𝑏𝑐 / 𝑎𝐶)
1614, 15sylan9eq 2797 . . . . 5 ((𝑎 = 𝑐𝑏 = 𝑑) → 𝐶 = 𝑑 / 𝑏𝑐 / 𝑎𝐶)
178, 9, 12, 13, 16cbvmpo 7527 . . . 4 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶)
1817oveqi 7444 . . 3 (𝑎(𝑎𝐴, 𝑏𝐵𝐶)𝑏) = (𝑎(𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶)𝑏)
19 eqidd 2738 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → (𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶) = (𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶))
20 equcom 2017 . . . . . . . 8 (𝑎 = 𝑐𝑐 = 𝑎)
21 equcom 2017 . . . . . . . 8 (𝑏 = 𝑑𝑑 = 𝑏)
2220, 21anbi12i 628 . . . . . . 7 ((𝑎 = 𝑐𝑏 = 𝑑) ↔ (𝑐 = 𝑎𝑑 = 𝑏))
2322, 16sylbir 235 . . . . . 6 ((𝑐 = 𝑎𝑑 = 𝑏) → 𝐶 = 𝑑 / 𝑏𝑐 / 𝑎𝐶)
2423eqcomd 2743 . . . . 5 ((𝑐 = 𝑎𝑑 = 𝑏) → 𝑑 / 𝑏𝑐 / 𝑎𝐶 = 𝐶)
2524adantl 481 . . . 4 (((𝜑𝑎𝐴𝑏𝐵) ∧ (𝑐 = 𝑎𝑑 = 𝑏)) → 𝑑 / 𝑏𝑐 / 𝑎𝐶 = 𝐶)
26 simp2 1138 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝑎𝐴)
27 simp3 1139 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝑏𝐵)
28 fvmpopr2d.3 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
2919, 25, 26, 27, 28ovmpod 7585 . . 3 ((𝜑𝑎𝐴𝑏𝐵) → (𝑎(𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶)𝑏) = 𝐶)
3018, 29eqtrid 2789 . 2 ((𝜑𝑎𝐴𝑏𝐵) → (𝑎(𝑎𝐴, 𝑏𝐵𝐶)𝑏) = 𝐶)
317, 30eqtr3d 2779 1 ((𝜑𝑎𝐴𝑏𝐵) → (𝐹𝑃) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  csb 3899  cop 4632  cfv 6561  (class class class)co 7431  cmpo 7433
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  mpomulcn  24891  mnringmulrcld  44247
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