Theorem erov2 8828

Description: The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

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)))

Assertion

Ref	Expression
erov2	⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ([𝑃] ∼ ⨣ [𝑄] ∼ ) = [(𝑃 + 𝑄)] ∼ )

Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   𝑃,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑋,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   + ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   ∼ ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐽,𝑝,𝑞,𝑥,𝑦,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑄,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧

Allowed substitution hints:   ⨣ (𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑈(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝐽(𝑢,𝑡,𝑠,𝑟)   𝑋(𝑥,𝑦)

Proof of Theorem erov2

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 8826	1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ([𝑃] ∼ ⨣ [𝑄] ∼ ) = [(𝑃 + 𝑄)] ∼ )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∃wrex 3066   ⊆ wss 3945   class class class wbr 5142   × cxp 5670  ⟶wf 6538  (class class class)co 7414  {coprab 7415   Er wer 8715  [cec 8716   / cqs 8717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-er 8718  df-ec 8720  df-qs 8724

This theorem is referenced by: (None)