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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 7844 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ([𝑃] ∼ ⨣ [𝑄] ∼ ) = [(𝑃 + 𝑄)] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 × cxp 5112 ⟶wf 5884 (class class class)co 6650 {coprab 6651 Er wer 7739 [cec 7740 / cqs 7741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-ec 7744 df-qs 7748 |
This theorem is referenced by: (None) |
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