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Theorem rngqiprngimfv 21308
Description: The value of the function 𝐹 at an element of (the base set of) a non-unital ring. (Contributed by AV, 24-Feb-2025.)
Hypotheses
Ref Expression
rng2idlring.r (𝜑𝑅 ∈ Rng)
rng2idlring.i (𝜑𝐼 ∈ (2Ideal‘𝑅))
rng2idlring.j 𝐽 = (𝑅s 𝐼)
rng2idlring.u (𝜑𝐽 ∈ Ring)
rng2idlring.b 𝐵 = (Base‘𝑅)
rng2idlring.t · = (.r𝑅)
rng2idlring.1 1 = (1r𝐽)
rngqiprngim.g = (𝑅 ~QG 𝐼)
rngqiprngim.q 𝑄 = (𝑅 /s )
rngqiprngim.c 𝐶 = (Base‘𝑄)
rngqiprngim.p 𝑃 = (𝑄 ×s 𝐽)
rngqiprngim.f 𝐹 = (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)
Assertion
Ref Expression
rngqiprngimfv ((𝜑𝐴𝐵) → (𝐹𝐴) = ⟨[𝐴] , ( 1 · 𝐴)⟩)
Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑥,𝐵   𝜑,𝑥   𝑥,𝐴   𝑥,   𝑥, 1   𝑥, ·
Allowed substitution hints:   𝑃(𝑥)   𝑄(𝑥)   𝑅(𝑥)   𝐹(𝑥)   𝐽(𝑥)

Proof of Theorem rngqiprngimfv
StepHypRef Expression
1 rngqiprngim.f . . 3 𝐹 = (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)
21a1i 11 . 2 ((𝜑𝐴𝐵) → 𝐹 = (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩))
3 eceq1 8784 . . . 4 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
4 oveq2 7439 . . . 4 (𝑥 = 𝐴 → ( 1 · 𝑥) = ( 1 · 𝐴))
53, 4opeq12d 4881 . . 3 (𝑥 = 𝐴 → ⟨[𝑥] , ( 1 · 𝑥)⟩ = ⟨[𝐴] , ( 1 · 𝐴)⟩)
65adantl 481 . 2 (((𝜑𝐴𝐵) ∧ 𝑥 = 𝐴) → ⟨[𝑥] , ( 1 · 𝑥)⟩ = ⟨[𝐴] , ( 1 · 𝐴)⟩)
7 simpr 484 . 2 ((𝜑𝐴𝐵) → 𝐴𝐵)
8 opex 5469 . . 3 ⟨[𝐴] , ( 1 · 𝐴)⟩ ∈ V
98a1i 11 . 2 ((𝜑𝐴𝐵) → ⟨[𝐴] , ( 1 · 𝐴)⟩ ∈ V)
102, 6, 7, 9fvmptd 7023 1 ((𝜑𝐴𝐵) → (𝐹𝐴) = ⟨[𝐴] , ( 1 · 𝐴)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cmpt 5225  cfv 6561  (class class class)co 7431  [cec 8743  Basecbs 17247  s cress 17274  .rcmulr 17298   /s cqus 17550   ×s cxps 17551   ~QG cqg 19140  Rngcrng 20149  1rcur 20178  Ringcrg 20230  2Idealc2idl 21259
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-in 3958  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-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-ec 8747
This theorem is referenced by:  rngqiprngghm  21309  rngqiprngimf1  21310  rngqiprngimfo  21311  rngqiprnglin  21312  rngqiprngfu  21327
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