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Mirrors > Home > MPE Home > Th. List > rngqiprngimfv | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
rng2idlring.r | ⊢ (𝜑 → 𝑅 ∈ Rng) |
rng2idlring.i | ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) |
rng2idlring.j | ⊢ 𝐽 = (𝑅 ↾s 𝐼) |
rng2idlring.u | ⊢ (𝜑 → 𝐽 ∈ Ring) |
rng2idlring.b | ⊢ 𝐵 = (Base‘𝑅) |
rng2idlring.t | ⊢ · = (.r‘𝑅) |
rng2idlring.1 | ⊢ 1 = (1r‘𝐽) |
rngqiprngim.g | ⊢ ∼ = (𝑅 ~QG 𝐼) |
rngqiprngim.q | ⊢ 𝑄 = (𝑅 /s ∼ ) |
rngqiprngim.c | ⊢ 𝐶 = (Base‘𝑄) |
rngqiprngim.p | ⊢ 𝑃 = (𝑄 ×s 𝐽) |
rngqiprngim.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) |
Ref | Expression |
---|---|
rngqiprngimfv | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = 〈[𝐴] ∼ , ( 1 · 𝐴)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngqiprngim.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)) |
3 | eceq1 8783 | . . . 4 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ ) | |
4 | oveq2 7439 | . . . 4 ⊢ (𝑥 = 𝐴 → ( 1 · 𝑥) = ( 1 · 𝐴)) | |
5 | 3, 4 | opeq12d 4886 | . . 3 ⊢ (𝑥 = 𝐴 → 〈[𝑥] ∼ , ( 1 · 𝑥)〉 = 〈[𝐴] ∼ , ( 1 · 𝐴)〉) |
6 | 5 | adantl 481 | . 2 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 = 𝐴) → 〈[𝑥] ∼ , ( 1 · 𝑥)〉 = 〈[𝐴] ∼ , ( 1 · 𝐴)〉) |
7 | simpr 484 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
8 | opex 5475 | . . 3 ⊢ 〈[𝐴] ∼ , ( 1 · 𝐴)〉 ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 〈[𝐴] ∼ , ( 1 · 𝐴)〉 ∈ V) |
10 | 2, 6, 7, 9 | fvmptd 7023 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = 〈[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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