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Theorem fvmpopr2d 7290
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 7138 . . 3 (𝑎(𝑎𝐴, 𝑏𝐵𝐶)𝑏) = ((𝑎𝐴, 𝑏𝐵𝐶)‘⟨𝑎, 𝑏⟩)
2 fvmpopr2d.1 . . . . 5 (𝜑𝐹 = (𝑎𝐴, 𝑏𝐵𝐶))
323ad2ant1 1130 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝐹 = (𝑎𝐴, 𝑏𝐵𝐶))
4 fvmpopr2d.2 . . . . 5 (𝜑𝑃 = ⟨𝑎, 𝑏⟩)
543ad2ant1 1130 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝑃 = ⟨𝑎, 𝑏⟩)
63, 5fveq12d 6652 . . 3 ((𝜑𝑎𝐴𝑏𝐵) → (𝐹𝑃) = ((𝑎𝐴, 𝑏𝐵𝐶)‘⟨𝑎, 𝑏⟩))
71, 6eqtr4id 2852 . 2 ((𝜑𝑎𝐴𝑏𝐵) → (𝑎(𝑎𝐴, 𝑏𝐵𝐶)𝑏) = (𝐹𝑃))
8 nfcv 2955 . . . . 5 𝑐𝐶
9 nfcv 2955 . . . . 5 𝑑𝐶
10 nfcv 2955 . . . . . 6 𝑎𝑑
11 nfcsb1v 3852 . . . . . 6 𝑎𝑐 / 𝑎𝐶
1210, 11nfcsbw 3854 . . . . 5 𝑎𝑑 / 𝑏𝑐 / 𝑎𝐶
13 nfcsb1v 3852 . . . . 5 𝑏𝑑 / 𝑏𝑐 / 𝑎𝐶
14 csbeq1a 3842 . . . . . 6 (𝑎 = 𝑐𝐶 = 𝑐 / 𝑎𝐶)
15 csbeq1a 3842 . . . . . 6 (𝑏 = 𝑑𝑐 / 𝑎𝐶 = 𝑑 / 𝑏𝑐 / 𝑎𝐶)
1614, 15sylan9eq 2853 . . . . 5 ((𝑎 = 𝑐𝑏 = 𝑑) → 𝐶 = 𝑑 / 𝑏𝑐 / 𝑎𝐶)
178, 9, 12, 13, 16cbvmpo 7227 . . . 4 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶)
1817oveqi 7148 . . 3 (𝑎(𝑎𝐴, 𝑏𝐵𝐶)𝑏) = (𝑎(𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶)𝑏)
19 eqidd 2799 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → (𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶) = (𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶))
20 equcom 2025 . . . . . . . 8 (𝑎 = 𝑐𝑐 = 𝑎)
21 equcom 2025 . . . . . . . 8 (𝑏 = 𝑑𝑑 = 𝑏)
2220, 21anbi12i 629 . . . . . . 7 ((𝑎 = 𝑐𝑏 = 𝑑) ↔ (𝑐 = 𝑎𝑑 = 𝑏))
2322, 16sylbir 238 . . . . . 6 ((𝑐 = 𝑎𝑑 = 𝑏) → 𝐶 = 𝑑 / 𝑏𝑐 / 𝑎𝐶)
2423eqcomd 2804 . . . . 5 ((𝑐 = 𝑎𝑑 = 𝑏) → 𝑑 / 𝑏𝑐 / 𝑎𝐶 = 𝐶)
2524adantl 485 . . . 4 (((𝜑𝑎𝐴𝑏𝐵) ∧ (𝑐 = 𝑎𝑑 = 𝑏)) → 𝑑 / 𝑏𝑐 / 𝑎𝐶 = 𝐶)
26 simp2 1134 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝑎𝐴)
27 simp3 1135 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝑏𝐵)
28 fvmpopr2d.3 . . . 4 ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
2919, 25, 26, 27, 28ovmpod 7281 . . 3 ((𝜑𝑎𝐴𝑏𝐵) → (𝑎(𝑐𝐴, 𝑑𝐵𝑑 / 𝑏𝑐 / 𝑎𝐶)𝑏) = 𝐶)
3018, 29syl5eq 2845 . 2 ((𝜑𝑎𝐴𝑏𝐵) → (𝑎(𝑎𝐴, 𝑏𝐵𝐶)𝑏) = 𝐶)
317, 30eqtr3d 2835 1 ((𝜑𝑎𝐴𝑏𝐵) → (𝐹𝑃) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  csb 3828  cop 4531  cfv 6324  (class class class)co 7135  cmpo 7137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140
This theorem is referenced by:  mnringmulrcld  40931
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