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Theorem qusaddvallem 16258
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 6239 . . . . . 6 (Base‘𝑅) ∈ V
62, 5syl6eqel 2738 . . . . 5 (𝜑𝑉 ∈ V)
7 erex 7811 . . . . 5 ( Er 𝑉 → (𝑉 ∈ V → ∈ V))
84, 6, 7sylc 65 . . . 4 (𝜑 ∈ V)
9 qusaddf.z . . . 4 (𝜑𝑅𝑍)
101, 2, 3, 8, 9quslem 16250 . . 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 16256 . . 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 16236 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
1643ad2ant1 1102 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → Er 𝑉)
1763ad2ant1 1102 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → 𝑉 ∈ V)
1816, 17, 3divsfval 16254 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹𝑋) = [𝑋] )
1916, 17, 3divsfval 16254 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹𝑌) = [𝑌] )
2018, 19oveq12d 6708 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = ([𝑋] [𝑌] ))
2116, 17, 3divsfval 16254 . 2 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹‘(𝑋 · 𝑌)) = [(𝑋 · 𝑌)] )
2215, 20, 213eqtr3d 2693 1 ((𝜑𝑋𝑉𝑌𝑉) → ([𝑋] [𝑌] ) = [(𝑋 · 𝑌)] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cop 4216   ciun 4552   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690   Er wer 7784  [cec 7785   / cqs 7786  Basecbs 15904   /s cqus 16212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-ov 6693  df-er 7787  df-ec 7789  df-qs 7793
This theorem is referenced by:  qusaddval  16260  qusmulval  16262
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