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Mirrors > Home > ILE Home > Th. List > eroprf2 | GIF version |
Description: Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
eropr2.1 | ⊢ 𝐽 = (𝐴 / ∼ ) |
eropr2.2 | ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑥 = [𝑝] ∼ ∧ 𝑦 = [𝑞] ∼ ) ∧ 𝑧 = [(𝑝 + 𝑞)] ∼ )} |
eropr2.3 | ⊢ (𝜑 → ∼ ∈ 𝑋) |
eropr2.4 | ⊢ (𝜑 → ∼ Er 𝑈) |
eropr2.5 | ⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
eropr2.6 | ⊢ (𝜑 → + :(𝐴 × 𝐴)⟶𝐴) |
eropr2.7 | ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴))) → ((𝑟 ∼ 𝑠 ∧ 𝑡 ∼ 𝑢) → (𝑟 + 𝑡) ∼ (𝑠 + 𝑢))) |
Ref | Expression |
---|---|
eroprf2 | ⊢ (𝜑 → ⨣ :(𝐽 × 𝐽)⟶𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eropr2.1 | . 2 ⊢ 𝐽 = (𝐴 / ∼ ) | |
2 | eropr2.3 | . 2 ⊢ (𝜑 → ∼ ∈ 𝑋) | |
3 | eropr2.4 | . 2 ⊢ (𝜑 → ∼ Er 𝑈) | |
4 | eropr2.5 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝑈) | |
5 | eropr2.6 | . 2 ⊢ (𝜑 → + :(𝐴 × 𝐴)⟶𝐴) | |
6 | eropr2.7 | . 2 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴))) → ((𝑟 ∼ 𝑠 ∧ 𝑡 ∼ 𝑢) → (𝑟 + 𝑡) ∼ (𝑠 + 𝑢))) | |
7 | eropr2.2 | . 2 ⊢ ⨣ = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((𝑥 = [𝑝] ∼ ∧ 𝑦 = [𝑞] ∼ ) ∧ 𝑧 = [(𝑝 + 𝑞)] ∼ )} | |
8 | 1, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 2, 2, 1 | eroprf 6522 | 1 ⊢ (𝜑 → ⨣ :(𝐽 × 𝐽)⟶𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 class class class wbr 3929 × cxp 4537 ⟶wf 5119 (class class class)co 5774 {coprab 5775 Er wer 6426 [cec 6427 / cqs 6428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-er 6429 df-ec 6431 df-qs 6435 |
This theorem is referenced by: (None) |
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