Theorem aprprop 14542

Description: If two structures have the same ring components (properties), df-apr 14531 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 14286	. . . . . . . 8 ⊢ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)
11	10	biimpi 120	. . . . . . 7 ⊢ (𝐾 ∈ Ring → 𝐿 ∈ Ring)
12	2, 7, 8, 11	grpsubpropdg 13862	. . . . . 6 ⊢ (𝐾 ∈ Ring → (-g‘𝐾) = (-g‘𝐿))
13	12	oveqd 6075	. . . . 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 6086	. . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
17	14, 2, 16, 8, 11	unitpropdg 14396	. . . . 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 4181	. 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 14531	. . 3 ⊢ #r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑟) ∧ 𝑦 ∈ (Base‘𝑟)) ∧ (𝑥(-g‘𝑟)𝑦) ∈ (Unit‘𝑟))})
22		fveq2 5675	. . . . . . 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 5675	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (-g‘𝑟) = (-g‘𝐾))
27	26	oveqd 6075	. . . . . 6 ⊢ (𝑟 = 𝐾 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝐾)𝑦))
28		fveq2 5675	. . . . . 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 4181	. . 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 13358	. . . . . 6 ⊢ Base Fn V
34		funfvex 5692	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐾 ∈ dom Base) → (Base‘𝐾) ∈ V)
35	34	funfni 5463	. . . . . 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 4870	. . . 4 ⊢ (𝐾 ∈ Ring → ((Base‘𝐾) × (Base‘𝐾)) ∈ V)
38		opabssxp 4829	. . . . 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 4255	. . 3 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))} ∈ V)
41	21, 31, 32, 40	fvmptd3 5776	. 2 ⊢ (𝐾 ∈ Ring → (#r‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ (𝑥(-g‘𝐾)𝑦) ∈ (Unit‘𝐾))})
42		fveq2 5675	. . . . . . 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 5675	. . . . . . 7 ⊢ (𝑟 = 𝐿 → (-g‘𝑟) = (-g‘𝐿))
47	46	oveqd 6075	. . . . . 6 ⊢ (𝑟 = 𝐿 → (𝑥(-g‘𝑟)𝑦) = (𝑥(-g‘𝐿)𝑦))
48		fveq2 5675	. . . . . 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 4181	. . 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 4870	. . . 4 ⊢ (𝐾 ∈ Ring → ((Base‘𝐿) × (Base‘𝐿)) ∈ V)
55		opabssxp 4829	. . . . 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 4255	. . 3 ⊢ (𝐾 ∈ Ring → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) ∧ (𝑥(-g‘𝐿)𝑦) ∈ (Unit‘𝐿))} ∈ V)
58	21, 51, 52, 57	fvmptd3 5776	. 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 3214  {copab 4175   × cxp 4752   Fn wfn 5352  ‘cfv 5357  (class class class)co 6058  Basecbs 13299  +_gcplusg 13377  ._rcmulr 13378  -_gcsg 13760  Ringcrg 14242  Unitcui 14334  #_rcapr 14530

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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9258  df-2 9316  df-3 9317  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306  df-plusg 13390  df-mulr 13391  df-0g 13558  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-grp 13761  df-minusg 13762  df-sbg 13763  df-cmn 14042  df-abl 14043  df-mgp 14163  df-ur 14206  df-srg 14210  df-ring 14244  df-oppr 14314  df-dvdsr 14336  df-unit 14337  df-apr 14531

This theorem is referenced by:  drngprop  14558