Theorem qusaddvallem 17627

Description: Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
qusaddf.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusaddf.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusaddf.r	⊢ (𝜑 → ∼ Er 𝑉)
qusaddf.z	⊢ (𝜑 → 𝑅 ∈ 𝑍)
qusaddf.e	⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
qusaddf.c	⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
qusaddflem.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
qusaddflem.g	⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})

Assertion

Ref	Expression
qusaddvallem	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([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

Step	Hyp	Ref	Expression
1		qusaddf.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
2		qusaddf.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		qusaddflem.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
4		qusaddf.r	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑉)
5		fvex 6898	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
6	2, 5	eqeltrdi 2873	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
7		erex 8725	. . . . 5 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
8	4, 6, 7	sylc 66	. . . 4 ⊢ (𝜑 → ∼ ∈ V)
9		qusaddf.z	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
10	1, 2, 3, 8, 9	quslem 17619	. . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ ))
11		qusaddf.c	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
12		qusaddf.e	. . . 4 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
13	4, 6, 3, 11, 12	ercpbl 17625	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
14		qusaddflem.g	. . 3 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
15	10, 13, 14	imasaddvallem 17605	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))
16	4	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∼ Er 𝑉)
17	6	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑉 ∈ V)
18	16, 17, 3	divsfval 17623	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑋) = [𝑋] ∼ )
19	16, 17, 3	divsfval 17623	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑌) = [𝑌] ∼ )
20	18, 19	oveq12d 7437	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = ([𝑋] ∼ ∙ [𝑌] ∼ ))
21	16, 17, 3	divsfval 17623	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘(𝑋 · 𝑌)) = [(𝑋 · 𝑌)] ∼ )
22	15, 20, 21	3eqtr3d 2808	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  ⟨cop 4597  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194  ‘cfv 6540  (class class class)co 7419   Er wer 8697  [cec 8698   / cqs 8699  Basecbs 17291   /_s cqus 17581

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-ov 7422  df-er 8700  df-ec 8702  df-qs 8706

This theorem is used by:  qusaddval  17629  qusmulval  17631