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Theorem qusaddflem 16813
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 6676 . . . . 5 (Base‘𝑅) ∈ V
62, 5syl6eqel 2918 . . . 4 (𝜑𝑉 ∈ V)
7 erex 8302 . . . 4 ( Er 𝑉 → (𝑉 ∈ V → ∈ V))
84, 6, 7sylc 65 . . 3 (𝜑 ∈ V)
9 qusaddf.z . . 3 (𝜑𝑅𝑍)
101, 2, 3, 8, 9quslem 16804 . 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 16810 . 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 16791 1 (𝜑 :((𝑉 / ) × (𝑉 / ))⟶(𝑉 / ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557  cop 4563   ciun 4910   class class class wbr 5057  cmpt 5137   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145   Er wer 8275  [cec 8276   / cqs 8277  Basecbs 16471   /s cqus 16766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-ov 7148  df-er 8278  df-ec 8280  df-qs 8284
This theorem is referenced by:  qusaddf  16815  qusmulf  16817
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