Theorem qusval 13620

Description: Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusval.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusval.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusval.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
qusval.e	⊢ (𝜑 → ∼ ∈ 𝑊)
qusval.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)

Assertion

Ref	Expression
qusval	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))

Distinct variable groups:   𝑥, ∼   𝜑,𝑥   𝑥,𝑅   𝑥,𝑉

Allowed substitution hints:   𝑈(𝑥)   𝐹(𝑥)   𝑊(𝑥)   𝑍(𝑥)

Proof of Theorem qusval

Dummy variables 𝑒 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusval.u	. 2 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
2		df-qus 13600	. . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
3	2	a1i 9	. . 3 ⊢ (𝜑 → /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)))
4		simprl 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑟 = 𝑅)
5	4	fveq2d 5679	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = (Base‘𝑅))
6		qusval.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
7	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑉 = (Base‘𝑅))
8	5, 7	eqtr4d 2270	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉)
9		eceq2 6817	. . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ )
10	9	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ )
11	8, 10	mpteq12dv 4197	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ))
12		qusval.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
13	11, 12	eqtr4di 2285	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
14	13, 4	oveq12d 6076	. . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = (𝐹 “s 𝑅))
15		qusval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
16	15	elexd 2829	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
17		qusval.e	. . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊)
18	17	elexd 2829	. . 3 ⊢ (𝜑 → ∼ ∈ V)
19		basfn 13355	. . . . . . . 8 ⊢ Base Fn V
20		funfvex 5692	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
21	20	funfni 5463	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
22	19, 16, 21	sylancr 414	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
23	6, 22	eqeltrd 2311	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ V)
24	23	mptexd 5918	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) ∈ V)
25	12, 24	eqeltrid 2321	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
26		imasex 13602	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑅 ∈ 𝑍) → (𝐹 “s 𝑅) ∈ V)
27	25, 15, 26	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V)
28	3, 14, 16, 18, 27	ovmpod 6189	. 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅))
29	1, 28	eqtrd 2267	1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ↦ cmpt 4176   Fn wfn 5352  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060  [cec 6778  Basecbs 13296   “_s cimas 13596   /_s cqus 13597

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-ec 6782  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mulr 13388  df-iimas 13599  df-qus 13600

This theorem is referenced by:  qusin  13623  qusbas  13624  qusaddval  13632  qusaddf  13633  qusmulval  13634  qusmulf  13635  qusgrp2  13914  qusrng  14186  qusring2  14294