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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 6490 | 1 ⊢ (𝜑 → ⨣ :(𝐽 × 𝐽)⟶𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 ⊆ wss 3041 class class class wbr 3899 × cxp 4507 ⟶wf 5089 (class class class)co 5742 {coprab 5743 Er wer 6394 [cec 6395 / cqs 6396 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-er 6397 df-ec 6399 df-qs 6403 |
This theorem is referenced by: (None) |
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