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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 6199 | 1 ⊢ (𝜑 → ⨣ :(𝐽 × 𝐽)⟶𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 ⊆ wss 2917 class class class wbr 3764 × cxp 4343 ⟶wf 4898 (class class class)co 5512 {coprab 5513 Er wer 6103 [cec 6104 / cqs 6105 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: (None) |
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