Theorem aprval 14574

Description: Expand Definition df-apr 14573. (Contributed by Jim Kingdon, 17-Feb-2025.)

Hypotheses

Ref	Expression
aprval.b	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
aprval.ap	⊢ (𝜑 → # = (#r‘𝑅))
aprval.s	⊢ (𝜑 → − = (-g‘𝑅))
aprval.u	⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
aprval.r	⊢ (𝜑 → 𝑅 ∈ Ring)
aprval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
aprval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
aprval	⊢ (𝜑 → (𝑋 # 𝑌 ↔ (𝑋 − 𝑌) ∈ 𝑈))

Proof of Theorem aprval

Dummy variables 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4129	. . 3 ⊢ (𝑋 # 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ # )
2		aprval.ap	. . . . . 6 ⊢ (𝜑 → # = (#r‘𝑅))
3		df-apr 14573	. . . . . . 7 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
4		fveq2 5693	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5	4	eleq2d 2308	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
6	4	eleq2d 2308	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑦 ∈ (Base‘𝑟) ↔ 𝑦 ∈ (Base‘𝑅)))
7	5, 6	anbi12d 477	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
8		fveq2 5693	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (-g‘𝑟) = (-g‘𝑅))
9	8	oveqd 6096	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝑅)𝑦))
10		fveq2 5693	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
11	9, 10	eleq12d 2309	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟) ↔ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅)))
12	7, 11	anbi12d 477	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟)) ↔ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))))
13	12	opabbidv 4195	. . . . . . 7 ⊢ (𝑟 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
14		aprval.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
15	14	elexd 2835	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ V)
16		basfn 13394	. . . . . . . . . 10 ⊢ Base Fn V
17		funfvex 5710	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
18	17	funfni 5481	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
19	16, 15, 18	sylancr 418	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
20		xpexg 4887	. . . . . . . . 9 ⊢ (((Base‘𝑅) ∈ V ∧ (Base‘𝑅) ∈ V) → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
21	19, 19, 20	syl2anc 415	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
22		opabssxp 4847	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ⊆ ((Base‘𝑅) × (Base‘𝑅))
23	22	a1i 9	. . . . . . . 8 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
24	21, 23	ssexd 4271	. . . . . . 7 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ∈ V)
25	3, 13, 15, 24	fvmptd3 5796	. . . . . 6 ⊢ (𝜑 → (#r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
26	2, 25	eqtrd 2271	. . . . 5 ⊢ (𝜑 → # = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))})
27	26	eleq2d 2308	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ # ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))}))
28		aprval.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
29		aprval.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
30	28, 29	eleqtrd 2317	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅))
31		aprval.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
32	31, 29	eleqtrd 2317	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑅))
33		oveq12 6088	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥(-g‘𝑅)𝑦) = (𝑋(-g‘𝑅)𝑌))
34	33	eleq1d 2307	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅) ↔ (𝑋(-g‘𝑅)𝑌) ∈ (Unit‘𝑅)))
35	34	opelopab2a 4405	. . . . 5 ⊢ ((𝑋 ∈ (Base‘𝑅) ∧ 𝑌 ∈ (Base‘𝑅)) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ↔ (𝑋(-g‘𝑅)𝑌) ∈ (Unit‘𝑅)))
36	30, 32, 35	syl2anc 415	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅))} ↔ (𝑋(-g‘𝑅)𝑌) ∈ (Unit‘𝑅)))
37	27, 36	bitrd 188	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ # ↔ (𝑋(-g‘𝑅)𝑌) ∈ (Unit‘𝑅)))
38	1, 37	bitrid 192	. 2 ⊢ (𝜑 → (𝑋 # 𝑌 ↔ (𝑋(-g‘𝑅)𝑌) ∈ (Unit‘𝑅)))
39		aprval.s	. . . 4 ⊢ (𝜑 → − = (-g‘𝑅))
40	39	oveqd 6096	. . 3 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(-g‘𝑅)𝑌))
41		aprval.u	. . 3 ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
42	40, 41	eleq12d 2309	. 2 ⊢ (𝜑 → ((𝑋 − 𝑌) ∈ 𝑈 ↔ (𝑋(-g‘𝑅)𝑌) ∈ (Unit‘𝑅)))
43	38, 42	bitr4d 191	1 ⊢ (𝜑 → (𝑋 # 𝑌 ↔ (𝑋 − 𝑌) ∈ 𝑈))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  ⟨cop 3711   class class class wbr 4128  {copab 4189   × cxp 4770   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  -_gcsg 13790  Ringcrg 14283  Unitcui 14376  #_rcapr 14572

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-apr 14573

This theorem is referenced by:  aprunit  14575  aprirr  14578  aprsym  14579  aprcotr  14580  aprnzr  14582  aprlring  14583  opprdrng  14603