Theorem pzriprnglem11 21824

Description: Lemma 11 for pzriprng 21830: 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 8748	. 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 21815	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
7	6	eqcomi 2772	. . . . 5 ⊢ (ℤ × ℤ) = (Base‘𝑅)
8	7	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (ℤ × ℤ) = (Base‘𝑅))
9		ovexd 7475	. . . . 5 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (𝑅 ~QG 𝐼) ∈ V)
10	2, 9	eqeltrid 2868	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ∼ ∈ V)
11	5	pzriprnglem1 21814	. . . . 5 ⊢ 𝑅 ∈ Rng
12	11	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → 𝑅 ∈ Rng)
13	4, 8, 10, 12	qusbas 17743	. . 3 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ((ℤ × ℤ) / ∼ ) = (Base‘𝑄))
14	2, 13	ax-mp 5	. 2 ⊢ ((ℤ × ℤ) / ∼ ) = (Base‘𝑄)
15		nfcv 2926	. . . 4 ⊢ Ⅎ𝑠{𝑒 ∣ 𝑒 = [𝑝] ∼ }
16		nfcv 2926	. . . 4 ⊢ Ⅎ𝑟{𝑒 ∣ 𝑒 = [𝑝] ∼ }
17		nfcv 2926	. . . 4 ⊢ Ⅎ𝑝{𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
18		eceq1 8782	. . . . . 6 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → [𝑝] ∼ = [⟨𝑠, 𝑟⟩] ∼ )
19	18	eqeq2d 2774	. . . . 5 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → (𝑒 = [𝑝] ∼ ↔ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ ))
20	19	abbidv 2829	. . . 4 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → {𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ })
21	15, 16, 17, 20	iunxpf 5831	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
22		iunab 5013	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ }
23		iuncom 4962	. . . 4 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
24		df-sn 4589	. . . . . . . . 9 ⊢ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
25	24	eqcomi 2772	. . . . . . . 8 ⊢ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ }
26	25	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℤ → {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ })
27	26	iuneq2i 4976	. . . . . 6 ⊢ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ }
28		simpr 490	. . . . . . . . . . . 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 21823	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℤ ∧ 𝑟 ∈ ℤ) → [⟨𝑠, 𝑟⟩] ∼ = (ℤ × {𝑟}))
33	32	ancoms 464	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) → [⟨𝑠, 𝑟⟩] ∼ = (ℤ × {𝑟}))
34	33	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) ∧ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ) → [⟨𝑠, 𝑟⟩] ∼ = (ℤ × {𝑟}))
35	28, 34	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) ∧ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ) → 𝑝 = (ℤ × {𝑟}))
36	35	ex 418	. . . . . . . . . 10 ⊢ ((𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑝 = [⟨𝑠, 𝑟⟩] ∼ → 𝑝 = (ℤ × {𝑟})))
37	36	rexlimdva 3167	. . . . . . . . 9 ⊢ (𝑟 ∈ ℤ → (∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ → 𝑝 = (ℤ × {𝑟})))
38		0zd 12729	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 0 ∈ ℤ)
39		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 𝑝 = (ℤ × {𝑟}))
40		opeq1 4843	. . . . . . . . . . . . 13 ⊢ (𝑠 = 0 → ⟨𝑠, 𝑟⟩ = ⟨0, 𝑟⟩)
41	40	eceq1d 8783	. . . . . . . . . . . 12 ⊢ (𝑠 = 0 → [⟨𝑠, 𝑟⟩] ∼ = [⟨0, 𝑟⟩] ∼ )
42	39, 41	eqeqan12d 2777	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) ∧ 𝑠 = 0) → (𝑝 = [⟨𝑠, 𝑟⟩] ∼ ↔ (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ ))
43		0zd 12729	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℤ → 0 ∈ ℤ)
44	5, 29, 30, 31, 2	pzriprnglem10 21823	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ 𝑟 ∈ ℤ) → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
45	43, 44	mpancom 701	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℤ → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
46	45	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℤ → (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ )
47	46	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ )
48	38, 42, 47	rspcedvd 3582	. . . . . . . . . 10 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ )
49	48	ex 418	. . . . . . . . 9 ⊢ (𝑟 ∈ ℤ → (𝑝 = (ℤ × {𝑟}) → ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ))
50	37, 49	impbid 215	. . . . . . . 8 ⊢ (𝑟 ∈ ℤ → (∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ ↔ 𝑝 = (ℤ × {𝑟})))
51	50	abbidv 2829	. . . . . . 7 ⊢ (𝑟 ∈ ℤ → {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})})
52		iunsn 5027	. . . . . . 7 ⊢ ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ }
53		df-sn 4589	. . . . . . 7 ⊢ {(ℤ × {𝑟})} = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})}
54	51, 52, 53	3eqtr4g 2823	. . . . . 6 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
55	27, 54	eqtrid 2810	. . . . 5 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
56	55	iuneq2i 4976	. . . 4 ⊢ ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
57	23, 56	eqtri 2786	. . 3 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
58	21, 22, 57	3eqtr3i 2794	. 2 ⊢ {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
59	1, 14, 58	3eqtr3i 2794	1 ⊢ (Base‘𝑄) = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454  {csn 4588  ⟨cop 4838  ∪ ciun 4954   × cxp 5653  ‘cfv 6555  (class class class)co 7440  [cec 8740   / cqs 8741  0cc0 11224  ℤcz 12717  Basecbs 17413   ↾_s cress 17434   /_s cqus 17703   ×_s cxps 17704   ~_QG cqg 19358  Rngcrng 20401  1_rcur 20434  ℤ_ringczring 21779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-addf 11303

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-er 8742  df-ec 8744  df-qs 8748  df-map 8874  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-inf 9459  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-starv 17469  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-unif 17477  df-hom 17478  df-cco 17479  df-0g 17638  df-prds 17644  df-imas 17706  df-qus 17707  df-xps 17708  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-grp 19173  df-minusg 19174  df-subg 19359  df-eqg 19361  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-ring 20488  df-cring 20489  df-subrng 20825  df-subrg 20849  df-cnfld 21706  df-zring 21780

This theorem is used by:  pzriprnglem12  21825  pzriprnglem13  21826  pzriprnglem14  21827