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Theorem rngqiprngimfv 21326
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 8783 . . . 4 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
4 oveq2 7439 . . . 4 (𝑥 = 𝐴 → ( 1 · 𝑥) = ( 1 · 𝐴))
53, 4opeq12d 4886 . . 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 5475 . . 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 1537  wcel 2106  Vcvv 3478  cop 4637  cmpt 5231  cfv 6563  (class class class)co 7431  [cec 8742  Basecbs 17245  s cress 17274  .rcmulr 17299   /s cqus 17552   ×s cxps 17553   ~QG cqg 19153  Rngcrng 20170  1rcur 20199  Ringcrg 20251  2Idealc2idl 21277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-ec 8746
This theorem is referenced by:  rngqiprngghm  21327  rngqiprngimf1  21328  rngqiprngimfo  21329  rngqiprnglin  21330  rngqiprngfu  21345
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