Theorem pzriprnglem11 21502

Description: Lemma 11 for pzriprng 21508: The base set of the quotient of 𝑅 and 𝐽. (Contributed by AV, 22-Mar-2025.)

Hypotheses

Ref	Expression
pzriprng.r	⊢ 𝑅 = (ℤring ×s ℤring)
pzriprng.i	⊢ 𝐼 = (ℤ × {0})
pzriprng.j	⊢ 𝐽 = (𝑅 ↾s 𝐼)
pzriprng.1	⊢ 1 = (1r‘𝐽)
pzriprng.g	⊢ ∼ = (𝑅 ~QG 𝐼)
pzriprng.q	⊢ 𝑄 = (𝑅 /s ∼ )

Assertion

Ref	Expression
pzriprnglem11	⊢ (Base‘𝑄) = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}

Distinct variable group:   ∼ ,𝑟

Allowed substitution hints:   𝑄(𝑟)   𝑅(𝑟)   1 (𝑟)   𝐼(𝑟)   𝐽(𝑟)

Proof of Theorem pzriprnglem11

Dummy variables 𝑒 𝑝 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-qs 8751	. 2 ⊢ ((ℤ × ℤ) / ∼ ) = {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ }
2		pzriprng.g	. . 3 ⊢ ∼ = (𝑅 ~QG 𝐼)
3		pzriprng.q	. . . . 5 ⊢ 𝑄 = (𝑅 /s ∼ )
4	3	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → 𝑄 = (𝑅 /s ∼ ))
5		pzriprng.r	. . . . . . 7 ⊢ 𝑅 = (ℤring ×s ℤring)
6	5	pzriprnglem2 21493	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
7	6	eqcomi 2746	. . . . 5 ⊢ (ℤ × ℤ) = (Base‘𝑅)
8	7	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (ℤ × ℤ) = (Base‘𝑅))
9		ovexd 7466	. . . . 5 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (𝑅 ~QG 𝐼) ∈ V)
10	2, 9	eqeltrid 2845	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ∼ ∈ V)
11	5	pzriprnglem1 21492	. . . . 5 ⊢ 𝑅 ∈ Rng
12	11	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → 𝑅 ∈ Rng)
13	4, 8, 10, 12	qusbas 17590	. . 3 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ((ℤ × ℤ) / ∼ ) = (Base‘𝑄))
14	2, 13	ax-mp 5	. 2 ⊢ ((ℤ × ℤ) / ∼ ) = (Base‘𝑄)
15		nfcv 2905	. . . 4 ⊢ Ⅎ𝑠{𝑒 ∣ 𝑒 = [𝑝] ∼ }
16		nfcv 2905	. . . 4 ⊢ Ⅎ𝑟{𝑒 ∣ 𝑒 = [𝑝] ∼ }
17		nfcv 2905	. . . 4 ⊢ Ⅎ𝑝{𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
18		eceq1 8784	. . . . . 6 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → [𝑝] ∼ = [⟨𝑠, 𝑟⟩] ∼ )
19	18	eqeq2d 2748	. . . . 5 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → (𝑒 = [𝑝] ∼ ↔ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ ))
20	19	abbidv 2808	. . . 4 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → {𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ })
21	15, 16, 17, 20	iunxpf 5859	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
22		iunab 5051	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ }
23		iuncom 4999	. . . 4 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
24		df-sn 4627	. . . . . . . . 9 ⊢ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
25	24	eqcomi 2746	. . . . . . . 8 ⊢ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ }
26	25	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℤ → {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ })
27	26	iuneq2i 5013	. . . . . 6 ⊢ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ }
28		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) ∧ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ) → 𝑝 = [⟨𝑠, 𝑟⟩] ∼ )
29		pzriprng.i	. . . . . . . . . . . . . . 15 ⊢ 𝐼 = (ℤ × {0})
30		pzriprng.j	. . . . . . . . . . . . . . 15 ⊢ 𝐽 = (𝑅 ↾s 𝐼)
31		pzriprng.1	. . . . . . . . . . . . . . 15 ⊢ 1 = (1r‘𝐽)
32	5, 29, 30, 31, 2	pzriprnglem10 21501	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℤ ∧ 𝑟 ∈ ℤ) → [⟨𝑠, 𝑟⟩] ∼ = (ℤ × {𝑟}))
33	32	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) → [⟨𝑠, 𝑟⟩] ∼ = (ℤ × {𝑟}))
34	33	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) ∧ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ) → [⟨𝑠, 𝑟⟩] ∼ = (ℤ × {𝑟}))
35	28, 34	eqtrd 2777	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) ∧ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ) → 𝑝 = (ℤ × {𝑟}))
36	35	ex 412	. . . . . . . . . 10 ⊢ ((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑝 = [⟨𝑠, 𝑟⟩] ∼ → 𝑝 = (ℤ × {𝑟})))
37	36	rexlimdva 3155	. . . . . . . . 9 ⊢ (𝑟 ∈ ℤ → (∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ → 𝑝 = (ℤ × {𝑟})))
38		0zd 12625	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 0 ∈ ℤ)
39		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 𝑝 = (ℤ × {𝑟}))
40		opeq1 4873	. . . . . . . . . . . . 13 ⊢ (𝑠 = 0 → ⟨𝑠, 𝑟⟩ = ⟨0, 𝑟⟩)
41	40	eceq1d 8785	. . . . . . . . . . . 12 ⊢ (𝑠 = 0 → [⟨𝑠, 𝑟⟩] ∼ = [⟨0, 𝑟⟩] ∼ )
42	39, 41	eqeqan12d 2751	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) ∧ 𝑠 = 0) → (𝑝 = [⟨𝑠, 𝑟⟩] ∼ ↔ (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ ))
43		0zd 12625	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℤ → 0 ∈ ℤ)
44	5, 29, 30, 31, 2	pzriprnglem10 21501	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ 𝑟 ∈ ℤ) → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
45	43, 44	mpancom 688	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℤ → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
46	45	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℤ → (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ )
47	46	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ )
48	38, 42, 47	rspcedvd 3624	. . . . . . . . . 10 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ )
49	48	ex 412	. . . . . . . . 9 ⊢ (𝑟 ∈ ℤ → (𝑝 = (ℤ × {𝑟}) → ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ))
50	37, 49	impbid 212	. . . . . . . 8 ⊢ (𝑟 ∈ ℤ → (∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ↔ 𝑝 = (ℤ × {𝑟})))
51	50	abbidv 2808	. . . . . . 7 ⊢ (𝑟 ∈ ℤ → {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})})
52		iunsn 5066	. . . . . . 7 ⊢ ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ }
53		df-sn 4627	. . . . . . 7 ⊢ {(ℤ × {𝑟})} = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})}
54	51, 52, 53	3eqtr4g 2802	. . . . . 6 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
55	27, 54	eqtrid 2789	. . . . 5 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
56	55	iuneq2i 5013	. . . 4 ⊢ ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
57	23, 56	eqtri 2765	. . 3 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
58	21, 22, 57	3eqtr3i 2773	. 2 ⊢ {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
59	1, 14, 58	3eqtr3i 2773	1 ⊢ (Base‘𝑄) = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070  Vcvv 3480  {csn 4626  ⟨cop 4632  ∪ ciun 4991   × cxp 5683  ‘cfv 6561  (class class class)co 7431  [cec 8743   / cqs 8744  0cc0 11155  ℤcz 12613  Basecbs 17247   ↾_s cress 17274   /_s cqus 17550   ×_s cxps 17551   ~_QG cqg 19140  Rngcrng 20149  1_rcur 20178  ℤ_ringczring 21457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-prds 17492  df-imas 17553  df-qus 17554  df-xps 17555  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-eqg 19143  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-subrng 20546  df-subrg 20570  df-cnfld 21365  df-zring 21458

This theorem is referenced by:  pzriprnglem12  21503  pzriprnglem13  21504  pzriprnglem14  21505