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Mirrors > Home > MPE Home > Th. List > erov2 | Structured version Visualization version GIF version |
Description: The value 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 |
---|---|
erov2 | ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ([𝑃] ∼ ⨣ [𝑄] ∼ ) = [(𝑃 + 𝑄)] ∼ ) |
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 | erov 8396 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ([𝑃] ∼ ⨣ [𝑄] ∼ ) = [(𝑃 + 𝑄)] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ⊆ wss 3938 class class class wbr 5068 × cxp 5555 ⟶wf 6353 (class class class)co 7158 {coprab 7159 Er wer 8288 [cec 8289 / cqs 8290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-ec 8293 df-qs 8297 |
This theorem is referenced by: (None) |
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