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Theorem qusaddflem 16193
Description: The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
qusaddf.u (𝜑𝑈 = (𝑅 /s ))
qusaddf.v (𝜑𝑉 = (Base‘𝑅))
qusaddf.r (𝜑 Er 𝑉)
qusaddf.z (𝜑𝑅𝑍)
qusaddf.e (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))
qusaddf.c ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
qusaddflem.f 𝐹 = (𝑥𝑉 ↦ [𝑥] )
qusaddflem.g (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (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
StepHypRef Expression
1 qusaddf.u . . 3 (𝜑𝑈 = (𝑅 /s ))
2 qusaddf.v . . 3 (𝜑𝑉 = (Base‘𝑅))
3 qusaddflem.f . . 3 𝐹 = (𝑥𝑉 ↦ [𝑥] )
4 qusaddf.r . . . 4 (𝜑 Er 𝑉)
5 fvex 6188 . . . . 5 (Base‘𝑅) ∈ V
62, 5syl6eqel 2707 . . . 4 (𝜑𝑉 ∈ V)
7 erex 7751 . . . 4 ( Er 𝑉 → (𝑉 ∈ V → ∈ V))
84, 6, 7sylc 65 . . 3 (𝜑 ∈ V)
9 qusaddf.z . . 3 (𝜑𝑅𝑍)
101, 2, 3, 8, 9quslem 16184 . 2 (𝜑𝐹:𝑉onto→(𝑉 / ))
11 qusaddf.c . . 3 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
12 qusaddf.e . . 3 (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))
134, 6, 3, 11, 12ercpbl 16190 . 2 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
14 qusaddflem.g . 2 (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
1510, 13, 14, 11imasaddflem 16171 1 (𝜑 :((𝑉 / ) × (𝑉 / ))⟶(𝑉 / ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  {csn 4168  cop 4174   ciun 4511   class class class wbr 4644  cmpt 4720   × cxp 5102  wf 5872  cfv 5876  (class class class)co 6635   Er wer 7724  [cec 7725   / cqs 7726  Basecbs 15838   /s cqus 16146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-ov 6638  df-er 7727  df-ec 7729  df-qs 7733
This theorem is referenced by:  qusaddf  16195  qusmulf  16197
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