Theorem qusaddflem 17477

Description: The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
qusaddf.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusaddf.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusaddf.r	⊢ (𝜑 → ∼ Er 𝑉)
qusaddf.z	⊢ (𝜑 → 𝑅 ∈ 𝑍)
qusaddf.e	⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
qusaddf.c	⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
qusaddflem.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
qusaddflem.g	⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})

Assertion

Ref	Expression
qusaddflem	⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))

Distinct variable groups:   𝑎,𝑏,𝑝,𝑞,𝑥, ∼   𝐹,𝑎,𝑏,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞,𝑥   𝑉,𝑎,𝑏,𝑝,𝑞,𝑥   𝑅,𝑝,𝑞,𝑥   · ,𝑝,𝑞,𝑥   ∙ ,𝑎,𝑏,𝑝,𝑞

Allowed substitution hints:   𝑅(𝑎,𝑏)   ∙ (𝑥)   · (𝑎,𝑏)   𝑈(𝑥,𝑞,𝑝,𝑎,𝑏)   𝐹(𝑥)   𝑍(𝑥,𝑞,𝑝,𝑎,𝑏)

Proof of Theorem qusaddflem

Step	Hyp	Ref	Expression
1		qusaddf.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
2		qusaddf.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		qusaddflem.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
4		qusaddf.r	. . . 4 ⊢ (𝜑 → ∼ Er 𝑉)
5		fvex 6888	. . . . 5 ⊢ (Base‘𝑅) ∈ V
6	2, 5	eqeltrdi 2840	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
7		erex 8707	. . . 4 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
8	4, 6, 7	sylc 65	. . 3 ⊢ (𝜑 → ∼ ∈ V)
9		qusaddf.z	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
10	1, 2, 3, 8, 9	quslem 17468	. 2 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ ))
11		qusaddf.c	. . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
12		qusaddf.e	. . 3 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
13	4, 6, 3, 11, 12	ercpbl 17474	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
14		qusaddflem.g	. 2 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
15	10, 13, 14, 11	imasaddflem 17455	1 ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3470  {csn 4619  ⟨cop 4625  ∪ ciun 4987   class class class wbr 5138   ↦ cmpt 5221   × cxp 5664  ⟶wf 6525  ‘cfv 6529  (class class class)co 7390   Er wer 8680  [cec 8681   / cqs 8682  Basecbs 17123   /_s cqus 17430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-fo 6535  df-fv 6537  df-ov 7393  df-er 8683  df-ec 8685  df-qs 8689

This theorem is referenced by:  qusaddf  17479  qusmulf  17481