Theorem pzriprnglem11 21797

Description: Lemma 11 for pzriprng 21803: 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 8723	. 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 21788	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
7	6	eqcomi 2769	. . . . 5 ⊢ (ℤ × ℤ) = (Base‘𝑅)
8	7	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (ℤ × ℤ) = (Base‘𝑅))
9		ovexd 7455	. . . . 5 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → (𝑅 ~QG 𝐼) ∈ V)
10	2, 9	eqeltrid 2864	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ∼ ∈ V)
11	5	pzriprnglem1 21787	. . . . 5 ⊢ 𝑅 ∈ Rng
12	11	a1i 11	. . . 4 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → 𝑅 ∈ Rng)
13	4, 8, 10, 12	qusbas 17717	. . 3 ⊢ ( ∼ = (𝑅 ~QG 𝐼) → ((ℤ × ℤ) / ∼ ) = (Base‘𝑄))
14	2, 13	ax-mp 5	. 2 ⊢ ((ℤ × ℤ) / ∼ ) = (Base‘𝑄)
15		nfcv 2922	. . . 4 ⊢ Ⅎ𝑠{𝑒 ∣ 𝑒 = [𝑝] ∼ }
16		nfcv 2922	. . . 4 ⊢ Ⅎ𝑟{𝑒 ∣ 𝑒 = [𝑝] ∼ }
17		nfcv 2922	. . . 4 ⊢ Ⅎ𝑝{𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
18		eceq1 8757	. . . . . 6 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → [𝑝] ∼ = [⟨𝑠, 𝑟⟩] ∼ )
19	18	eqeq2d 2771	. . . . 5 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → (𝑒 = [𝑝] ∼ ↔ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ ))
20	19	abbidv 2826	. . . 4 ⊢ (𝑝 = ⟨𝑠, 𝑟⟩ → {𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ })
21	15, 16, 17, 20	iunxpf 5825	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
22		iunab 5009	. . 3 ⊢ ∪ 𝑝 ∈ (ℤ × ℤ){𝑒 ∣ 𝑒 = [𝑝] ∼ } = {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ }
23		iuncom 4958	. . . 4 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
24		df-sn 4584	. . . . . . . . 9 ⊢ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ }
25	24	eqcomi 2769	. . . . . . . 8 ⊢ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ }
26	25	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℤ → {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {[⟨𝑠, 𝑟⟩] ∼ })
27	26	iuneq2i 4972	. . . . . 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 21796	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . 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 3163	. . . . . . . . 9 ⊢ (𝑟 ∈ ℤ → (∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ → 𝑝 = (ℤ × {𝑟})))
38		0zd 12705	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 0 ∈ ℤ)
39		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) → 𝑝 = (ℤ × {𝑟}))
40		opeq1 4832	. . . . . . . . . . . . 13 ⊢ (𝑠 = 0 → ⟨𝑠, 𝑟⟩ = ⟨0, 𝑟⟩)
41	40	eceq1d 8758	. . . . . . . . . . . 12 ⊢ (𝑠 = 0 → [⟨𝑠, 𝑟⟩] ∼ = [⟨0, 𝑟⟩] ∼ )
42	39, 41	eqeqan12d 2774	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ ℤ ∧ 𝑝 = (ℤ × {𝑟})) ∧ 𝑠 = 0) → (𝑝 = [⟨𝑠, 𝑟⟩] ∼ ↔ (ℤ × {𝑟}) = [⟨0, 𝑟⟩] ∼ ))
43		0zd 12705	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℤ → 0 ∈ ℤ)
44	5, 29, 30, 31, 2	pzriprnglem10 21796	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ 𝑟 ∈ ℤ) → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
45	43, 44	mpancom 701	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℤ → [⟨0, 𝑟⟩] ∼ = (ℤ × {𝑟}))
46	45	eqcomd 2766	. . . . . . . . . . . 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 3578	. . . . . . . . . 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 2826	. . . . . . 7 ⊢ (𝑟 ∈ ℤ → {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})})
52		iunsn 5023	. . . . . . 7 ⊢ ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {𝑝 ∣ ∃𝑠 ∈ ℤ 𝑝 = [⟨𝑠, 𝑟⟩] ∼ }
53		df-sn 4584	. . . . . . 7 ⊢ {(ℤ × {𝑟})} = {𝑝 ∣ 𝑝 = (ℤ × {𝑟})}
54	51, 52, 53	3eqtr4g 2820	. . . . . 6 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {[⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
55	27, 54	eqtrid 2807	. . . . 5 ⊢ (𝑟 ∈ ℤ → ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = {(ℤ × {𝑟})})
56	55	iuneq2i 4972	. . . 4 ⊢ ∪ 𝑟 ∈ ℤ ∪ 𝑠 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
57	23, 56	eqtri 2783	. . 3 ⊢ ∪ 𝑠 ∈ ℤ ∪ 𝑟 ∈ ℤ {𝑒 ∣ 𝑒 = [⟨𝑠, 𝑟⟩] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
58	21, 22, 57	3eqtr3i 2791	. 2 ⊢ {𝑒 ∣ ∃𝑝 ∈ (ℤ × ℤ)𝑒 = [𝑝] ∼ } = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
59	1, 14, 58	3eqtr3i 2791	1 ⊢ (Base‘𝑄) = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450  {csn 4583  ⟨cop 4589  ∪ ciun 4950   × cxp 5648  ‘cfv 6538  (class class class)co 7420  [cec 8715   / cqs 8716  0cc0 11200  ℤcz 12693  Basecbs 17387   ↾_s cress 17408   /_s cqus 17677   ×_s cxps 17678   ~_QG cqg 19332  Rngcrng 20374  1_rcur 20407  ℤ_ringczring 21752

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-addf 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-ec 8719  df-qs 8723  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-0g 17612  df-prds 17618  df-imas 17680  df-qus 17681  df-xps 17682  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-subg 19333  df-eqg 19335  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-subrng 20798  df-subrg 20822  df-cnfld 21679  df-zring 21753

This theorem is used by:  pzriprnglem12  21798  pzriprnglem13  21799  pzriprnglem14  21800