Theorem qusaddflemg 12772

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 12534	. . . . . 6 ⊢ Base Fn V
6		qusaddf.z	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
7	6	elexd 2762	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
8		funfvex 5544	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
9	8	funfni 5328	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
10	5, 7, 9	sylancr 414	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
11	2, 10	eqeltrd 2264	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
12		erex 6573	. . . 4 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
13	4, 11, 12	sylc 62	. . 3 ⊢ (𝜑 → ∼ ∈ V)
14	1, 2, 3, 13, 6	quslem 12763	. 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 12769	. 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 12755	1 ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363   ∈ wcel 2158  Vcvv 2749  {csn 3604  ⟨cop 3607  ∪ ciun 3898   class class class wbr 4015   ↦ cmpt 4076   × cxp 4636   Fn wfn 5223  ⟶wf 5224  ‘cfv 5228  (class class class)co 5888   Er wer 6546  [cec 6547   / cqs 6548  Basecbs 12476   /_s cqus 12739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7916  ax-resscn 7917  ax-1re 7919  ax-addrcl 7922

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-er 6549  df-ec 6551  df-qs 6555  df-inn 8934  df-ndx 12479  df-slot 12480  df-base 12482

This theorem is referenced by:  qusaddf  12774  qusmulf  12776