Theorem qusaddvallemg 13136

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))⟩})
qusaddflemg.x	⊢ (𝜑 → · ∈ 𝑊)

Assertion

Ref	Expression
qusaddvallemg	⊢ ((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)   𝑊(𝑥,𝑞,𝑝,𝑎,𝑏)   𝑋(𝑎,𝑏)   𝑌(𝑎,𝑏)   𝑍(𝑥,𝑞,𝑝,𝑎,𝑏)

Proof of Theorem qusaddvallemg

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		qusaddf.z	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
6		basfn 12861	. . . . . . . 8 ⊢ Base Fn V
7		elex 2782	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑍 → 𝑅 ∈ V)
8		funfvex 5592	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
9	8	funfni 5375	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
10	6, 7, 9	sylancr 414	. . . . . . 7 ⊢ (𝑅 ∈ 𝑍 → (Base‘𝑅) ∈ V)
11	5, 10	syl 14	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
12	2, 11	eqeltrd 2281	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
13		erex 6643	. . . . 5 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
14	4, 12, 13	sylc 62	. . . 4 ⊢ (𝜑 → ∼ ∈ V)
15	1, 2, 3, 14, 5	quslem 13127	. . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ ))
16		qusaddf.c	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
17		qusaddf.e	. . . 4 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
18	4, 12, 3, 16, 17	ercpbl 13134	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
19		qusaddflem.g	. . 3 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
20		qusaddflemg.x	. . 3 ⊢ (𝜑 → · ∈ 𝑊)
21	15, 18, 19, 12, 20	imasaddvallemg 13118	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))
22	4	3ad2ant1 1020	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∼ Er 𝑉)
23	12	3ad2ant1 1020	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑉 ∈ V)
24		simp2 1000	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝑉)
25	22, 23, 3, 24	divsfvalg 13132	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑋) = [𝑋] ∼ )
26		simp3 1001	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉)
27	22, 23, 3, 26	divsfvalg 13132	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑌) = [𝑌] ∼ )
28	25, 27	oveq12d 5961	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = ([𝑋] ∼ ∙ [𝑌] ∼ ))
29	16	3ad2antl1 1161	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
30	29, 24, 26	caovcld 6099	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉)
31	22, 23, 3, 30	divsfvalg 13132	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘(𝑋 · 𝑌)) = [(𝑋 · 𝑌)] ∼ )
32	21, 28, 31	3eqtr3d 2245	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  Vcvv 2771  {csn 3632  ⟨cop 3635  ∪ ciun 3926   class class class wbr 4043   ↦ cmpt 4104   Fn wfn 5265  ‘cfv 5270  (class class class)co 5943   Er wer 6616  [cec 6617   / cqs 6618  Basecbs 12803   /_s cqus 13103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-er 6619  df-ec 6621  df-qs 6625  df-inn 9036  df-ndx 12806  df-slot 12807  df-base 12809

This theorem is referenced by:  qusaddval  13138  qusmulval  13140