Theorem aprprop 14584

Description: If two structures have the same ring components (properties), df-apr 14573 generates the same relation for both of them. (Contributed by Jim Kingdon, 31-May-2026.)

Hypotheses

Ref	Expression
aprprop.b	⊢ (Base‘𝐾) = (Base‘𝐿)
aprprop.p	⊢ (+g‘𝐾) = (+g‘𝐿)
aprprop.m	⊢ (.r‘𝐾) = (.r‘𝐿)

Assertion

Ref	Expression
aprprop	⊢ (𝐾 ∈ Ring → (#r‘𝐾) = (#r‘𝐿))

Proof of Theorem aprprop

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

Step	Hyp	Ref	Expression
1		aprprop.b	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐿)
2	1	a1i 9	. . . . . 6 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘𝐿))
3	2	eleq2d 2308	. . . . 5 ⊢ (𝐾 ∈ Ring → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥 ∈ (Base‘𝐿)))
4	2	eleq2d 2308	. . . . 5 ⊢ (𝐾 ∈ Ring → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦 ∈ (Base‘𝐿)))
5	3, 4	anbi12d 477	. . . 4 ⊢ (𝐾 ∈ Ring → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿))))
6		aprprop.p	. . . . . . . 8 ⊢ (+g‘𝐾) = (+g‘𝐿)
7	6	a1i 9	. . . . . . 7 ⊢ (𝐾 ∈ Ring → (+g‘𝐾) = (+g‘𝐿))
8		id 19	. . . . . . 7 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ Ring)
9		aprprop.m	. . . . . . . . 9 ⊢ (.r‘𝐾) = (.r‘𝐿)
10	1, 6, 9	ringprop 14328	. . . . . . . 8 ⊢ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)
11	10	biimpi 120	. . . . . . 7 ⊢ (𝐾 ∈ Ring → 𝐿 ∈ Ring)
12	2, 7, 8, 11	grpsubpropdg 13892	. . . . . 6 ⊢ (𝐾 ∈ Ring → (-g‘𝐾) = (-g‘𝐿))
13	12	oveqd 6096	. . . . 5 ⊢ (𝐾 ∈ Ring → (𝑥(-g‘𝐾)𝑦) = (𝑥(-g‘𝐿)𝑦))
14		eqidd 2239	. . . . . 6 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘𝐾))
15	9	a1i 9	. . . . . . 7 ⊢ (𝐾 ∈ Ring → (.r‘𝐾) = (.r‘𝐿))
16	15	oveqdr 6107	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
17	14, 2, 16, 8, 11	unitpropdg 14438	. . . . 5 ⊢ (𝐾 ∈ Ring → (Unit‘𝐾) = (Unit‘𝐿))
18	13, 17	eleq12d 2309	. . . 4 ⊢ (𝐾 ∈ Ring → ((𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾) ↔ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿)))
19	5, 18	anbi12d 477	. . 3 ⊢ (𝐾 ∈ Ring → (((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾)) ↔ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))))
20	19	opabbidv 4195	. 2 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))})
21		df-apr 14573	. . 3 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
22		fveq2 5693	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (Base‘𝑟) = (Base‘𝐾))
23	22	eleq2d 2308	. . . . . 6 ⊢ (𝑟 = 𝐾 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝐾)))
24	22	eleq2d 2308	. . . . . 6 ⊢ (𝑟 = 𝐾 → (𝑦 ∈ (Base‘𝑟) ↔ 𝑦 ∈ (Base‘𝐾)))
25	23, 24	anbi12d 477	. . . . 5 ⊢ (𝑟 = 𝐾 → ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
26		fveq2 5693	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (-g‘𝑟) = (-g‘𝐾))
27	26	oveqd 6096	. . . . . 6 ⊢ (𝑟 = 𝐾 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝐾)𝑦))
28		fveq2 5693	. . . . . 6 ⊢ (𝑟 = 𝐾 → (Unit‘𝑟) = (Unit‘𝐾))
29	27, 28	eleq12d 2309	. . . . 5 ⊢ (𝑟 = 𝐾 → ((𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟) ↔ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾)))
30	25, 29	anbi12d 477	. . . 4 ⊢ (𝑟 = 𝐾 → (((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟)) ↔ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))))
31	30	opabbidv 4195	. . 3 ⊢ (𝑟 = 𝐾 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))})
32		elex 2833	. . 3 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ V)
33		basfn 13394	. . . . . 6 ⊢ Base Fn V
34		funfvex 5710	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
35	34	funfni 5481	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐾 ∈ V) → (Base‘𝐾) ∈ V)
36	33, 32, 35	sylancr 418	. . . . 5 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) ∈ V)
37	36, 36	xpexd 4888	. . . 4 ⊢ (𝐾 ∈ Ring → ((Base‘𝐾) × (Base‘𝐾)) ∈ V)
38		opabssxp 4847	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))} ⊆ ((Base‘𝐾) × (Base‘𝐾))
39	38	a1i 9	. . . 4 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))} ⊆ ((Base‘𝐾) × (Base‘𝐾)))
40	37, 39	ssexd 4271	. . 3 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))} ∈ V)
41	21, 31, 32, 40	fvmptd3 5796	. 2 ⊢ (𝐾 ∈ Ring → (#r‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))})
42		fveq2 5693	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (Base‘𝑟) = (Base‘𝐿))
43	42	eleq2d 2308	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝐿)))
44	42	eleq2d 2308	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑦 ∈ (Base‘𝑟) ↔ 𝑦 ∈ (Base‘𝐿)))
45	43, 44	anbi12d 477	. . . . 5 ⊢ (𝑟 = 𝐿 → ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿))))
46		fveq2 5693	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (-g‘𝑟) = (-g‘𝐿))
47	46	oveqd 6096	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝐿)𝑦))
48		fveq2 5693	. . . . . 6 ⊢ (𝑟 = 𝐿 → (Unit‘𝑟) = (Unit‘𝐿))
49	47, 48	eleq12d 2309	. . . . 5 ⊢ (𝑟 = 𝐿 → ((𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟) ↔ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿)))
50	45, 49	anbi12d 477	. . . 4 ⊢ (𝑟 = 𝐿 → (((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟)) ↔ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))))
51	50	opabbidv 4195	. . 3 ⊢ (𝑟 = 𝐿 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))})
52	11	elexd 2835	. . 3 ⊢ (𝐾 ∈ Ring → 𝐿 ∈ V)
53	1, 36	eqeltrrid 2326	. . . . 5 ⊢ (𝐾 ∈ Ring → (Base‘𝐿) ∈ V)
54	53, 53	xpexd 4888	. . . 4 ⊢ (𝐾 ∈ Ring → ((Base‘𝐿) × (Base‘𝐿)) ∈ V)
55		opabssxp 4847	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))} ⊆ ((Base‘𝐿) × (Base‘𝐿))
56	55	a1i 9	. . . 4 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))} ⊆ ((Base‘𝐿) × (Base‘𝐿)))
57	54, 56	ssexd 4271	. . 3 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))} ∈ V)
58	21, 51, 52, 57	fvmptd3 5796	. 2 ⊢ (𝐾 ∈ Ring → (#r‘𝐿) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))})
59	20, 41, 58	3eqtr4d 2281	1 ⊢ (𝐾 ∈ Ring → (#r‘𝐾) = (#r‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  {copab 4189   × cxp 4770   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  +_gcplusg 13414  ._rcmulr 13415  -_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-in1 623  ax-in2 624  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-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  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-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-tpos 6510  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-sbg 13793  df-cmn 14072  df-abl 14073  df-mgp 14201  df-ur 14246  df-srg 14251  df-ring 14285  df-oppr 14356  df-dvdsr 14378  df-unit 14379  df-apr 14573

This theorem is referenced by:  drngprop  14600