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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 8784 | . . . 4 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ ) | |
| 4 | oveq2 7439 | . . . 4 ⊢ (𝑥 = 𝐴 → ( 1 · 𝑥) = ( 1 · 𝐴)) | |
| 5 | 3, 4 | opeq12d 4881 | . . 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 5469 | . . 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 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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