Theorem pzriprnglem11 21433

Description: Lemma 11 for pzriprng 21439: 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 8654	. 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 21424	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
7	6	eqcomi 2738	. . . . 5 ⊢ (ℤ × ℤ) = (Base‘𝑅)
8	7	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (ℤ × ℤ) = (Base‘𝑅))
9		ovexd 7404	. . . . 5 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (𝑅 ~QG 𝐼) ∈ V)
10	2, 9	eqeltrid 2832	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ∼ ∈ V)
11	5	pzriprnglem1 21423	. . . . 5 ⊢ 𝑅 ∈ Rng
12	11	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → 𝑅 ∈ Rng)
13	4, 8, 10, 12	qusbas 17484	. . 3 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ((ℤ × ℤ) / ∼ ) = (Base‘𝑄))
14	2, 13	ax-mp 5	. 2 ⊢ ((ℤ × ℤ) / ∼ ) = (Base‘𝑄)
15		nfcv 2891	. . . 4 ⊢ Ⅎ𝑠{𝑒 ∣ 𝑒 = [𝑝] ∼ }
16		nfcv 2891	. . . 4 ⊢ Ⅎ𝑟{𝑒 ∣ 𝑒 = [𝑝] ∼ }
17		nfcv 2891	. . . 4 ⊢ Ⅎ𝑝{𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
18		eceq1 8687	. . . . . 6 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → [𝑝] ∼ = [⟨𝑠, 𝑟⟩] ∼ )
19	18	eqeq2d 2740	. . . . 5 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → (𝑒 = [𝑝] ∼ ↔ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ ))
20	19	abbidv 2795	. . . 4 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → {𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ })
21	15, 16, 17, 20	iunxpf 5802	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
22		iunab 5010	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ }
23		iuncom 4959	. . . 4 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
24		df-sn 4586	. . . . . . . . 9 ⊢ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
25	24	eqcomi 2738	. . . . . . . 8 ⊢ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ }
26	25	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℤ → {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ })
27	26	iuneq2i 4973	. . . . . 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 21432	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . 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 3134	. . . . . . . . 9 ⊢ (𝑟 ∈ ℤ → (∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ → 𝑝 = (ℤ × {𝑟})))
38		0zd 12517	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 0 ∈ ℤ)
39		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 𝑝 = (ℤ × {𝑟}))
40		opeq1 4833	. . . . . . . . . . . . 13 ⊢ (𝑠 = 0 → ⟨𝑠, 𝑟⟩ = ⟨0, 𝑟⟩)
41	40	eceq1d 8688	. . . . . . . . . . . 12 ⊢ (𝑠 = 0 → [⟨𝑠, 𝑟⟩] ∼ = [⟨0, 𝑟⟩] ∼ )
42	39, 41	eqeqan12d 2743	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) ∧ 𝑠 = 0) → (𝑝 = [⟨𝑠, 𝑟⟩] ∼ ↔ (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ ))
43		0zd 12517	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℤ → 0 ∈ ℤ)
44	5, 29, 30, 31, 2	pzriprnglem10 21432	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ 𝑟 ∈ ℤ) → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
45	43, 44	mpancom 688	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℤ → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
46	45	eqcomd 2735	. . . . . . . . . . . 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 3587	. . . . . . . . . 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 2795	. . . . . . 7 ⊢ (𝑟 ∈ ℤ → {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})})
52		iunsn 5025	. . . . . . 7 ⊢ ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ }
53		df-sn 4586	. . . . . . 7 ⊢ {(ℤ × {𝑟})} = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})}
54	51, 52, 53	3eqtr4g 2789	. . . . . 6 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
55	27, 54	eqtrid 2776	. . . . 5 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
56	55	iuneq2i 4973	. . . 4 ⊢ ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
57	23, 56	eqtri 2752	. . 3 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
58	21, 22, 57	3eqtr3i 2760	. 2 ⊢ {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
59	1, 14, 58	3eqtr3i 2760	1 ⊢ (Base‘𝑄) = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3444  {csn 4585  ⟨cop 4591  ∪ ciun 4951   × cxp 5629  ‘cfv 6499  (class class class)co 7369  [cec 8646   / cqs 8647  0cc0 11044  ℤcz 12505  Basecbs 17155   ↾_s cress 17176   /_s cqus 17444   ×_s cxps 17445   ~_QG cqg 19036  Rngcrng 20072  1_rcur 20101  ℤ_ringczring 21388

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-addf 11123

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-ec 8650  df-qs 8654  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-0g 17380  df-prds 17386  df-imas 17447  df-qus 17448  df-xps 17449  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-grp 18850  df-minusg 18851  df-subg 19037  df-eqg 19039  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-cring 20156  df-subrng 20466  df-subrg 20490  df-cnfld 21297  df-zring 21389

This theorem is referenced by:  pzriprnglem12  21434  pzriprnglem13  21435  pzriprnglem14  21436