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 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.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)

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 6671 . . . . 5 (Base‘𝑅) ∈ V
62, 5eqeltrdi 2860 . . . 4 (𝜑𝑉 ∈ V)
7 erex 8323 . . . 4 ( Er 𝑉 → (𝑉 ∈ V → ∈ V))
84, 6, 7sylc 65 . . 3 (𝜑 ∈ V)
9 qusaddf.z . . 3 (𝜑𝑅𝑍)
101, 2, 3, 8, 9quslem 16874 . 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 16880 . 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 16861 1 (𝜑 :((𝑉 / ) × (𝑉 / ))⟶(𝑉 / ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4522  ⟨cop 4528  ∪ ciun 4883   class class class wbr 5032   ↦ cmpt 5112   × cxp 5522  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   Er wer 8296  [cec 8297   / cqs 8298  Basecbs 16541   /s cqus 16836 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fo 6341  df-fv 6343  df-ov 7153  df-er 8299  df-ec 8301  df-qs 8305 This theorem is referenced by:  qusaddf  16885  qusmulf  16887
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