Theorem pzriprnglem11 21519

Description: Lemma 11 for pzriprng 21525: 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 8749	. 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 21510	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
7	6	eqcomi 2743	. . . . 5 ⊢ (ℤ × ℤ) = (Base‘𝑅)
8	7	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (ℤ × ℤ) = (Base‘𝑅))
9		ovexd 7465	. . . . 5 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (𝑅 ~QG 𝐼) ∈ V)
10	2, 9	eqeltrid 2842	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ∼ ∈ V)
11	5	pzriprnglem1 21509	. . . . 5 ⊢ 𝑅 ∈ Rng
12	11	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → 𝑅 ∈ Rng)
13	4, 8, 10, 12	qusbas 17591	. . 3 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ((ℤ × ℤ) / ∼ ) = (Base‘𝑄))
14	2, 13	ax-mp 5	. 2 ⊢ ((ℤ × ℤ) / ∼ ) = (Base‘𝑄)
15		nfcv 2902	. . . 4 ⊢ Ⅎ𝑠{𝑒 ∣ 𝑒 = [𝑝] ∼ }
16		nfcv 2902	. . . 4 ⊢ Ⅎ𝑟{𝑒 ∣ 𝑒 = [𝑝] ∼ }
17		nfcv 2902	. . . 4 ⊢ Ⅎ𝑝{𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
18		eceq1 8782	. . . . . 6 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → [𝑝] ∼ = [⟨𝑠, 𝑟⟩] ∼ )
19	18	eqeq2d 2745	. . . . 5 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → (𝑒 = [𝑝] ∼ ↔ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ ))
20	19	abbidv 2805	. . . 4 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → {𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ })
21	15, 16, 17, 20	iunxpf 5861	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
22		iunab 5055	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ }
23		iuncom 5003	. . . 4 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
24		df-sn 4631	. . . . . . . . 9 ⊢ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
25	24	eqcomi 2743	. . . . . . . 8 ⊢ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ }
26	25	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℤ → {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ })
27	26	iuneq2i 5017	. . . . . 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 21518	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . 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 3152	. . . . . . . . 9 ⊢ (𝑟 ∈ ℤ → (∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ → 𝑝 = (ℤ × {𝑟})))
38		0zd 12622	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 0 ∈ ℤ)
39		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 𝑝 = (ℤ × {𝑟}))
40		opeq1 4877	. . . . . . . . . . . . 13 ⊢ (𝑠 = 0 → ⟨𝑠, 𝑟⟩ = ⟨0, 𝑟⟩)
41	40	eceq1d 8783	. . . . . . . . . . . 12 ⊢ (𝑠 = 0 → [⟨𝑠, 𝑟⟩] ∼ = [⟨0, 𝑟⟩] ∼ )
42	39, 41	eqeqan12d 2748	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) ∧ 𝑠 = 0) → (𝑝 = [⟨𝑠, 𝑟⟩] ∼ ↔ (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ ))
43		0zd 12622	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℤ → 0 ∈ ℤ)
44	5, 29, 30, 31, 2	pzriprnglem10 21518	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ 𝑟 ∈ ℤ) → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
45	43, 44	mpancom 688	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℤ → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
46	45	eqcomd 2740	. . . . . . . . . . . 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 3623	. . . . . . . . . 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 2805	. . . . . . 7 ⊢ (𝑟 ∈ ℤ → {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})})
52		iunsn 5070	. . . . . . 7 ⊢ ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ }
53		df-sn 4631	. . . . . . 7 ⊢ {(ℤ × {𝑟})} = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})}
54	51, 52, 53	3eqtr4g 2799	. . . . . 6 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
55	27, 54	eqtrid 2786	. . . . 5 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
56	55	iuneq2i 5017	. . . 4 ⊢ ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
57	23, 56	eqtri 2762	. . 3 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
58	21, 22, 57	3eqtr3i 2770	. 2 ⊢ {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
59	1, 14, 58	3eqtr3i 2770	1 ⊢ (Base‘𝑄) = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}

Syntax hints:   ∧ wa 395   = wceq 1536   ∈ wcel 2105  {cab 2711  ∃wrex 3067  Vcvv 3477  {csn 4630  ⟨cop 4636  ∪ ciun 4995   × cxp 5686  ‘cfv 6562  (class class class)co 7430  [cec 8741   / cqs 8742  0cc0 11152  ℤcz 12610  Basecbs 17244   ↾_s cress 17273   /_s cqus 17551   ×_s cxps 17552   ~_QG cqg 19152  Rngcrng 20169  1_rcur 20198  ℤ_ringczring 21474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-addf 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-ec 8745  df-qs 8749  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  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 17487  df-prds 17493  df-imas 17554  df-qus 17555  df-xps 17556  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-subg 19153  df-eqg 19155  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-subrng 20562  df-subrg 20586  df-cnfld 21382  df-zring 21475

This theorem is referenced by:  pzriprnglem12  21520  pzriprnglem13  21521  pzriprnglem14  21522