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Theorem qusaddvallem 17596
Description: Value of an operation defined on a quotient structure. (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
qusaddvallem ((𝜑𝑋𝑉𝑌𝑉) → ([𝑋] [𝑌] ) = [(𝑋 · 𝑌)] )
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   𝑋,𝑝,𝑞,𝑥   ,𝑎,𝑏,𝑝,𝑞   𝑌,𝑝,𝑞,𝑥
Allowed substitution hints:   𝑅(𝑎,𝑏)   (𝑥)   · (𝑎,𝑏)   𝑈(𝑥,𝑞,𝑝,𝑎,𝑏)   𝐹(𝑥)   𝑋(𝑎,𝑏)   𝑌(𝑎,𝑏)   𝑍(𝑥,𝑞,𝑝,𝑎,𝑏)

Proof of Theorem qusaddvallem
StepHypRef Expression
1 qusaddf.u . . . 4 (𝜑𝑈 = (𝑅 /s ))
2 qusaddf.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
3 qusaddflem.f . . . 4 𝐹 = (𝑥𝑉 ↦ [𝑥] )
4 qusaddf.r . . . . 5 (𝜑 Er 𝑉)
5 fvex 6919 . . . . . 6 (Base‘𝑅) ∈ V
62, 5eqeltrdi 2849 . . . . 5 (𝜑𝑉 ∈ V)
7 erex 8769 . . . . 5 ( Er 𝑉 → (𝑉 ∈ V → ∈ V))
84, 6, 7sylc 65 . . . 4 (𝜑 ∈ V)
9 qusaddf.z . . . 4 (𝜑𝑅𝑍)
101, 2, 3, 8, 9quslem 17588 . . 3 (𝜑𝐹:𝑉onto→(𝑉 / ))
11 qusaddf.c . . . 4 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
12 qusaddf.e . . . 4 (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))
134, 6, 3, 11, 12ercpbl 17594 . . 3 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
14 qusaddflem.g . . 3 (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
1510, 13, 14imasaddvallem 17574 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
1643ad2ant1 1134 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → Er 𝑉)
1763ad2ant1 1134 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → 𝑉 ∈ V)
1816, 17, 3divsfval 17592 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹𝑋) = [𝑋] )
1916, 17, 3divsfval 17592 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹𝑌) = [𝑌] )
2018, 19oveq12d 7449 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = ([𝑋] [𝑌] ))
2116, 17, 3divsfval 17592 . 2 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹‘(𝑋 · 𝑌)) = [(𝑋 · 𝑌)] )
2215, 20, 213eqtr3d 2785 1 ((𝜑𝑋𝑉𝑌𝑉) → ([𝑋] [𝑌] ) = [(𝑋 · 𝑌)] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   ciun 4991   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431   Er wer 8742  [cec 8743   / cqs 8744  Basecbs 17247   /s cqus 17550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-er 8745  df-ec 8747  df-qs 8751
This theorem is referenced by:  qusaddval  17598  qusmulval  17600
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