Theorem qusval 17495

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 17462	. . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
3	2	a1i 11	. . 3 ⊢ (𝜑 → /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)))
4		simprl 768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑟 = 𝑅)
5	4	fveq2d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = (Base‘𝑅))
6		qusval.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
7	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → 𝑉 = (Base‘𝑅))
8	5, 7	eqtr4d 2774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (Base‘𝑟) = 𝑉)
9		eceq2 8749	. . . . . . 7 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ )
10	9	ad2antll 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → [𝑥]𝑒 = [𝑥] ∼ )
11	8, 10	mpteq12dv 5239	. . . . 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 2789	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑒 = ∼ )) → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = 𝐹)
14	13, 4	oveq12d 7430	. . 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 3494	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
17		qusval.e	. . . 4 ⊢ (𝜑 → ∼ ∈ 𝑊)
18	17	elexd 3494	. . 3 ⊢ (𝜑 → ∼ ∈ V)
19		ovexd 7447	. . 3 ⊢ (𝜑 → (𝐹 “s 𝑅) ∈ V)
20	3, 14, 16, 18, 19	ovmpod 7563	. 2 ⊢ (𝜑 → (𝑅 /s ∼ ) = (𝐹 “s 𝑅))
21	1, 20	eqtrd 2771	1 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ↦ cmpt 5231  ‘cfv 6543  (class class class)co 7412   ∈ cmpo 7414  [cec 8707  Basecbs 17151   “_s cimas 17457   /_s cqus 17458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-ec 8711  df-qus 17462

This theorem is referenced by:  qusin  17497  qusbas  17498  quss  17499  qusaddval  17506  qusaddf  17507  qusmulval  17508  qusmulf  17509  qusgrp2  18984  qusrng  20081  qusring2  20229  qustps  23546  qustgpopn  23944  qustgplem  23945  qustgphaus  23947  qusvsval  32903  quslmod  32909  quslmhm  32910