Theorem aprprop 14603

Description: If two structures have the same ring components (properties), df-apr 14592 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 14347	. . . . . . . 8 ⊢ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)
11	10	biimpi 120	. . . . . . 7 ⊢ (𝐾 ∈ Ring → 𝐿 ∈ Ring)
12	2, 7, 8, 11	grpsubpropdg 13911	. . . . . 6 ⊢ (𝐾 ∈ Ring → (-g‘𝐾) = (-g‘𝐿))
13	12	oveqd 6102	. . . . 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 6113	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
17	14, 2, 16, 8, 11	unitpropdg 14457	. . . . 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 4197	. 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 14592	. . 3 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
22		fveq2 5695	. . . . . . 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 5695	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (-g‘𝑟) = (-g‘𝐾))
27	26	oveqd 6102	. . . . . 6 ⊢ (𝑟 = 𝐾 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝐾)𝑦))
28		fveq2 5695	. . . . . 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 4197	. . 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 13413	. . . . . 6 ⊢ Base Fn V
34		funfvex 5712	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
35	34	funfni 5483	. . . . . 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 4890	. . . 4 ⊢ (𝐾 ∈ Ring → ((Base‘𝐾) × (Base‘𝐾)) ∈ V)
38		opabssxp 4849	. . . . 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 4273	. . 3 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))} ∈ V)
41	21, 31, 32, 40	fvmptd3 5799	. 2 ⊢ (𝐾 ∈ Ring → (#r‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))})
42		fveq2 5695	. . . . . . 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 5695	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (-g‘𝑟) = (-g‘𝐿))
47	46	oveqd 6102	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝐿)𝑦))
48		fveq2 5695	. . . . . 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 4197	. . 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 4890	. . . 4 ⊢ (𝐾 ∈ Ring → ((Base‘𝐿) × (Base‘𝐿)) ∈ V)
55		opabssxp 4849	. . . . 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 4273	. . 3 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))} ∈ V)
58	21, 51, 52, 57	fvmptd3 5799	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  {copab 4191   × cxp 4772   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  Basecbs 13354  +_gcplusg 13433  ._rcmulr 13434  -_gcsg 13809  Ringcrg 14302  Unitcui 14395  #_rcapr 14591

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-plusg 13446  df-mulr 13447  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-sbg 13812  df-cmn 14091  df-abl 14092  df-mgp 14220  df-ur 14265  df-srg 14270  df-ring 14304  df-oppr 14375  df-dvdsr 14397  df-unit 14398  df-apr 14592

This theorem is used by:  drngprop  14619