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 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.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)

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 6676 . . . . . 6 (Base‘𝑅) ∈ V
62, 5eqeltrdi 2860 . . . . 5 (𝜑𝑉 ∈ V)
7 erex 8329 . . . . 5 ( Er 𝑉 → (𝑉 ∈ V → ∈ V))
84, 6, 7sylc 65 . . . 4 (𝜑 ∈ V)
9 qusaddf.z . . . 4 (𝜑𝑅𝑍)
101, 2, 3, 8, 9quslem 16888 . . 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 16894 . . 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 16874 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))
1643ad2ant1 1130 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → Er 𝑉)
1763ad2ant1 1130 . . . 4 ((𝜑𝑋𝑉𝑌𝑉) → 𝑉 ∈ V)
1816, 17, 3divsfval 16892 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹𝑋) = [𝑋] )
1916, 17, 3divsfval 16892 . . 3 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹𝑌) = [𝑌] )
2018, 19oveq12d 7174 . 2 ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = ([𝑋] [𝑌] ))
2116, 17, 3divsfval 16892 . 2 ((𝜑𝑋𝑉𝑌𝑉) → (𝐹‘(𝑋 · 𝑌)) = [(𝑋 · 𝑌)] )
2215, 20, 213eqtr3d 2801 1 ((𝜑𝑋𝑉𝑌𝑉) → ([𝑋] [𝑌] ) = [(𝑋 · 𝑌)] )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4525  ⟨cop 4531  ∪ ciun 4886   class class class wbr 5036   ↦ cmpt 5116  ‘cfv 6340  (class class class)co 7156   Er wer 8302  [cec 8303   / cqs 8304  Basecbs 16555   /s cqus 16850 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 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 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 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fo 6346  df-fv 6348  df-ov 7159  df-er 8305  df-ec 8307  df-qs 8311 This theorem is referenced by:  qusaddval  16898  qusmulval  16900
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