Theorem qusaddflemg 13655

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))⟩})
qusaddflemg.x	⊢ (𝜑 → · ∈ 𝑊)

Assertion

Ref	Expression
qusaddflemg	⊢ (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)   𝑊(𝑥, 𝑞, 𝑝, 𝑎, 𝑏)   𝑍(𝑥, 𝑞, 𝑝, 𝑎, 𝑏)

Proof of Theorem qusaddflemg

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		basfn 13411	. . . . . 6 ⊢ Base Fn V
6		qusaddf.z	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
7	6	elexd 2835	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
8		funfvex 5712	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
9	8	funfni 5483	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
10	5, 7, 9	sylancr 418	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
11	2, 10	eqeltrd 2315	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
12		erex 6831	. . . 4 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
13	4, 11, 12	sylc 62	. . 3 ⊢ (𝜑 → ∼ ∈ V)
14	1, 2, 3, 13, 6	quslem 13645	. 2 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ ))
15		qusaddf.c	. . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
16		qusaddf.e	. . 3 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
17	4, 11, 3, 15, 16	ercpbl 13652	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
18		qusaddflem.g	. 2 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
19		qusaddflemg.x	. 2 ⊢ (𝜑 → · ∈ 𝑊)
20	14, 17, 18, 11, 19, 15	imasaddflemg 13637	1 ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3709  ⟨cop 3712  ∪ ciun 4012   class class class wbr 4130   ↦ cmpt 4192   × cxp 4772   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085   Er wer 6804  [cec 6805   / cqs 6806  Basecbs 13352   /_s cqus 13623

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-er 6807  df-ec 6809  df-qs 6813  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358

This theorem is used by:  qusaddf  13657  qusmulf  13659