Theorem eroprf2 8741

Description: Functionality 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
eroprf2	⊢ (𝜑 → ⨣ :(𝐽 × 𝐽)⟶𝐽)

Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   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

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 eroprf2

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 8739	1 ⊢ (𝜑 → ⨣ :(𝐽 × 𝐽)⟶𝐽)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897   class class class wbr 5089   × cxp 5612  ⟶wf 6477  (class class class)co 7346  {coprab 7347   Er wer 8619  [cec 8620   / cqs 8621

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-er 8622  df-ec 8624  df-qs 8628

This theorem is referenced by: (None)