Theorem aprprop 14461

Description: If two structures have the same ring components (properties), df-apr 14450 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 2304	. . . . 5 ⊢ (𝐾 ∈ Ring → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥 ∈ (Base‘𝐿)))
4	2	eleq2d 2304	. . . . 5 ⊢ (𝐾 ∈ Ring → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦 ∈ (Base‘𝐿)))
5	3, 4	anbi12d 473	. . . 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 14205	. . . . . . . 8 ⊢ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)
11	10	biimpi 120	. . . . . . 7 ⊢ (𝐾 ∈ Ring → 𝐿 ∈ Ring)
12	2, 7, 8, 11	grpsubpropdg 13838	. . . . . 6 ⊢ (𝐾 ∈ Ring → (-g‘𝐾) = (-g‘𝐿))
13	12	oveqd 6069	. . . . 5 ⊢ (𝐾 ∈ Ring → (𝑥(-g‘𝐾)𝑦) = (𝑥(-g‘𝐿)𝑦))
14		eqidd 2235	. . . . . 6 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘𝐾))
15	9	a1i 9	. . . . . . 7 ⊢ (𝐾 ∈ Ring → (.r‘𝐾) = (.r‘𝐿))
16	15	oveqdr 6080	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
17	14, 2, 16, 8, 11	unitpropdg 14315	. . . . 5 ⊢ (𝐾 ∈ Ring → (Unit‘𝐾) = (Unit‘𝐿))
18	13, 17	eleq12d 2305	. . . 4 ⊢ (𝐾 ∈ Ring → ((𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾) ↔ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿)))
19	5, 18	anbi12d 473	. . 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 4178	. 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 14450	. . 3 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
22		fveq2 5672	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (Base‘𝑟) = (Base‘𝐾))
23	22	eleq2d 2304	. . . . . 6 ⊢ (𝑟 = 𝐾 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝐾)))
24	22	eleq2d 2304	. . . . . 6 ⊢ (𝑟 = 𝐾 → (𝑦 ∈ (Base‘𝑟) ↔ 𝑦 ∈ (Base‘𝐾)))
25	23, 24	anbi12d 473	. . . . 5 ⊢ (𝑟 = 𝐾 → ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
26		fveq2 5672	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (-g‘𝑟) = (-g‘𝐾))
27	26	oveqd 6069	. . . . . 6 ⊢ (𝑟 = 𝐾 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝐾)𝑦))
28		fveq2 5672	. . . . . 6 ⊢ (𝑟 = 𝐾 → (Unit‘𝑟) = (Unit‘𝐾))
29	27, 28	eleq12d 2305	. . . . 5 ⊢ (𝑟 = 𝐾 → ((𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟) ↔ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾)))
30	25, 29	anbi12d 473	. . . 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 4178	. . 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 2827	. . 3 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ V)
33		basfn 13292	. . . . . 6 ⊢ Base Fn V
34		funfvex 5689	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
35	34	funfni 5460	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐾 ∈ V) → (Base‘𝐾) ∈ V)
36	33, 32, 35	sylancr 414	. . . . 5 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) ∈ V)
37	36, 36	xpexd 4867	. . . 4 ⊢ (𝐾 ∈ Ring → ((Base‘𝐾) × (Base‘𝐾)) ∈ V)
38		opabssxp 4826	. . . . 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 4252	. . 3 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))} ∈ V)
41	21, 31, 32, 40	fvmptd3 5773	. 2 ⊢ (𝐾 ∈ Ring → (#r‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))})
42		fveq2 5672	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (Base‘𝑟) = (Base‘𝐿))
43	42	eleq2d 2304	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝐿)))
44	42	eleq2d 2304	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑦 ∈ (Base‘𝑟) ↔ 𝑦 ∈ (Base‘𝐿)))
45	43, 44	anbi12d 473	. . . . 5 ⊢ (𝑟 = 𝐿 → ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿))))
46		fveq2 5672	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (-g‘𝑟) = (-g‘𝐿))
47	46	oveqd 6069	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝐿)𝑦))
48		fveq2 5672	. . . . . 6 ⊢ (𝑟 = 𝐿 → (Unit‘𝑟) = (Unit‘𝐿))
49	47, 48	eleq12d 2305	. . . . 5 ⊢ (𝑟 = 𝐿 → ((𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟) ↔ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿)))
50	45, 49	anbi12d 473	. . . 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 4178	. . 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 2829	. . 3 ⊢ (𝐾 ∈ Ring → 𝐿 ∈ V)
53	1, 36	eqeltrrid 2322	. . . . 5 ⊢ (𝐾 ∈ Ring → (Base‘𝐿) ∈ V)
54	53, 53	xpexd 4867	. . . 4 ⊢ (𝐾 ∈ Ring → ((Base‘𝐿) × (Base‘𝐿)) ∈ V)
55		opabssxp 4826	. . . . 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 4252	. . 3 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))} ∈ V)
58	21, 51, 52, 57	fvmptd3 5773	. 2 ⊢ (𝐾 ∈ Ring → (#r‘𝐿) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))})
59	20, 41, 58	3eqtr4d 2277	1 ⊢ (𝐾 ∈ Ring → (#r‘𝐾) = (#r‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3213  {copab 4172   × cxp 4749   Fn wfn 5349  ‘cfv 5354  (class class class)co 6052  Basecbs 13233  +_gcplusg 13311  ._rcmulr 13312  -_gcsg 13736  Ringcrg 14161  Unitcui 14253  #_rcapr 14449

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-lttrn 8246  ax-pre-ltadd 8248

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-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-tpos 6478  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-grp 13737  df-minusg 13738  df-sbg 13739  df-cmn 14024  df-abl 14025  df-mgp 14086  df-ur 14125  df-srg 14129  df-ring 14163  df-oppr 14233  df-dvdsr 14255  df-unit 14256  df-apr 14450

This theorem is referenced by:  drngprop  14477