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Theorem pzriprnglem10 21418

Description: Lemma 10 for pzriprng 21425: The equivalence classes of 𝑅 modulo 𝐽. (Contributed by AV, 22-Mar-2025.)

**Hypotheses**
Ref	Expression
pzriprng.r	⊢ 𝑅 = (ℤ_ring ×_s ℤ_ring)
pzriprng.i	⊢ 𝐼 = (ℤ × {0})
pzriprng.j	⊢ 𝐽 = (𝑅 ↾_s 𝐼)
pzriprng.1	⊢ 1 = (1_r‘𝐽)
pzriprng.g	⊢ ∼ = (𝑅 ~_QG 𝐼)

**Assertion**
Ref	Expression
pzriprnglem10	⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = (ℤ × {𝑌}))

Proof of Theorem pzriprnglem10 Dummy variables 𝑥 𝑦 𝑎 𝑏 𝑐 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pzriprng.r	. . . . 5 ⊢ 𝑅 = (ℤ_ring ×_s ℤ_ring)
2	1	pzriprnglem1 21409	. . . 4 ⊢ 𝑅 ∈ Rng
3		rnggrp 20100	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
4	2, 3	ax-mp 5	. . 3 ⊢ 𝑅 ∈ Grp
5		pzriprng.i	. . . . 5 ⊢ 𝐼 = (ℤ × {0})
6		0z 12597	. . . . . 6 ⊢ 0 ∈ ℤ
7		snssi 4807	. . . . . 6 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
8		xpss2 5692	. . . . . 6 ⊢ ({0} ⊆ ℤ → (ℤ × {0}) ⊆ (ℤ × ℤ))
9	6, 7, 8	mp2b 10	. . . . 5 ⊢ (ℤ × {0}) ⊆ (ℤ × ℤ)
10	5, 9	eqsstri 4007	. . . 4 ⊢ 𝐼 ⊆ (ℤ × ℤ)
11	10	a1i 11	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝐼 ⊆ (ℤ × ℤ))
12		opelxpi 5709	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ⟨𝑋, 𝑌⟩ ∈ (ℤ × ℤ))
13	1	pzriprnglem2 21410	. . . . 5 ⊢ (Base‘𝑅) = (ℤ × ℤ)
14	13	eqcomi 2734	. . . 4 ⊢ (ℤ × ℤ) = (Base‘𝑅)
15		pzriprng.g	. . . 4 ⊢ ∼ = (𝑅 ~_QG 𝐼)
16		eqid 2725	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
17	14, 15, 16	eqglact 19136	. . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ⊆ (ℤ × ℤ) ∧ ⟨𝑋, 𝑌⟩ ∈ (ℤ × ℤ)) → [⟨𝑋, 𝑌⟩] ∼ = ((𝑥 ∈ (ℤ × ℤ) ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) “ 𝐼))
18	4, 11, 12, 17	mp3an2i 1462	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = ((𝑥 ∈ (ℤ × ℤ) ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) “ 𝐼))
19	11	mptimass 6071	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((𝑥 ∈ (ℤ × ℤ) ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) “ 𝐼) = ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)))
20		eqid 2725	. . . . 5 ⊢ (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥))
21	20	rnmpt 5951	. . . 4 ⊢ ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = {𝑒 ∣ ∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)}
22	21	a1i 11	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = {𝑒 ∣ ∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)})
23	5	rexeqi 3314	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ ∃𝑥 ∈ (ℤ × {0})𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥))
24		oveq2 7423	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩))
25	24	eqeq2d 2736	. . . . . . 7 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)))
26	25	rexxp 5839	. . . . . 6 ⊢ (∃𝑥 ∈ (ℤ × {0})𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩))
27	23, 26	bitri 274	. . . . 5 ⊢ (∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩))
28	27	a1i 11	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)))
29	28	abbidv 2794	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ ∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)} = {𝑒 ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)})
30		c0ex 11236	. . . . . . . 8 ⊢ 0 ∈ V
31		opeq2 4870	. . . . . . . . . 10 ⊢ (𝑏 = 0 → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 0⟩)
32	31	oveq2d 7431	. . . . . . . . 9 ⊢ (𝑏 = 0 → (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩))
33	32	eqeq2d 2736	. . . . . . . 8 ⊢ (𝑏 = 0 → (𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) ↔ 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩)))
34	30, 33	rexsn 4682	. . . . . . 7 ⊢ (∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) ↔ 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩))
35		zringbas 21381	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤ_ring)
36		zringring 21377	. . . . . . . . . 10 ⊢ ℤ_ring ∈ Ring
37	36	a1i 11	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → ℤ_ring ∈ Ring)
38		simpll 765	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 𝑋 ∈ ℤ)
39		simplr 767	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 𝑌 ∈ ℤ)
40		simpr 483	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 𝑎 ∈ ℤ)
41		0zd 12598	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 0 ∈ ℤ)
42	38, 40	zaddcld 12698	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑋 + 𝑎) ∈ ℤ)
43		simpr 483	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑌 ∈ ℤ)
44		0zd 12598	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 0 ∈ ℤ)
45	43, 44	zaddcld 12698	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 + 0) ∈ ℤ)
46	45	adantr 479	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑌 + 0) ∈ ℤ)
47		zringplusg 21382	. . . . . . . . 9 ⊢ + = (+_g‘ℤ_ring)
48	1, 35, 35, 37, 37, 38, 39, 40, 41, 42, 46, 47, 47, 16	xpsadd 17553	. . . . . . . 8 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩) = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩)
49	48	eqeq2d 2736	. . . . . . 7 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩) ↔ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩))
50	34, 49	bitrid 282	. . . . . 6 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) ↔ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩))
51	50	rexbidva 3167	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) ↔ ∃𝑎 ∈ ℤ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩))
52	51	abbidv 2794	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)} = {𝑒 ∣ ∃𝑎 ∈ ℤ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩})
53		iunab 5049	. . . . . 6 ⊢ ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = {𝑒 ∣ ∃𝑎 ∈ ℤ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩}
54	53	eqcomi 2734	. . . . 5 ⊢ {𝑒 ∣ ∃𝑎 ∈ ℤ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩}
55	54	a1i 11	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ ∃𝑎 ∈ ℤ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩})
56		zcn 12591	. . . . . . . . . . 11 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℂ)
57	56	adantl 480	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑌 ∈ ℂ)
58	57	addridd 11442	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 + 0) = 𝑌)
59	58	opeq2d 4876	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
60	59	eqeq2d 2736	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩ ↔ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩))
61	60	abbidv 2794	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩})
62	61	iuneq2d 5020	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩})
63		df-sn 4625	. . . . . . . . 9 ⊢ {⟨(𝑋 + 𝑎), 𝑌⟩} = {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩}
64	63	eqcomi 2734	. . . . . . . 8 ⊢ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩} = {⟨(𝑋 + 𝑎), 𝑌⟩}
65	64	a1i 11	. . . . . . 7 ⊢ (𝑎 ∈ ℤ → {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩} = {⟨(𝑋 + 𝑎), 𝑌⟩})
66	65	iuneq2i 5012	. . . . . 6 ⊢ ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩} = ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩}
67	66	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩} = ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩})
68		velsn 4640	. . . . . . . . . 10 ⊢ (𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
69	68	rexbii 3084	. . . . . . . . 9 ⊢ (∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
70	42	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) → (𝑋 + 𝑎) ∈ ℤ)
71		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) ∧ 𝑏 = (𝑋 + 𝑎)) → 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
72		opeq1 4869	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑋 + 𝑎) → ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
73	72	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) ∧ 𝑏 = (𝑋 + 𝑎)) → ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
74	71, 73	eqeq12d 2741	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) ∧ 𝑏 = (𝑋 + 𝑎)) → (𝑦 = ⟨𝑏, 𝑌⟩ ↔ ⟨(𝑋 + 𝑎), 𝑌⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩))
75		eqidd 2726	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) → ⟨(𝑋 + 𝑎), 𝑌⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
76	70, 74, 75	rspcedvd 3604	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) → ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩)
77	76	ex 411	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩ → ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩))
78	77	rexlimdva 3145	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩ → ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩))
79		simpr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℤ)
80		simpll 765	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑋 ∈ ℤ)
81	79, 80	zsubcld 12699	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑏 − 𝑋) ∈ ℤ)
82	81	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → (𝑏 − 𝑋) ∈ ℤ)
83		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → 𝑦 = ⟨𝑏, 𝑌⟩)
84		oveq2 7423	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑏 − 𝑋) → (𝑋 + 𝑎) = (𝑋 + (𝑏 − 𝑋)))
85	84	adantl 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → (𝑋 + 𝑎) = (𝑋 + (𝑏 − 𝑋)))
86	85	opeq1d 4875	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → ⟨(𝑋 + 𝑎), 𝑌⟩ = ⟨(𝑋 + (𝑏 − 𝑋)), 𝑌⟩)
87	83, 86	eqeq12d 2741	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → (𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩ ↔ ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + (𝑏 − 𝑋)), 𝑌⟩))
88		zcn 12591	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ)
89	88	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑋 ∈ ℂ)
90	89	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑋 ∈ ℂ)
91		zcn 12591	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ)
92	91	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
93	90, 92	pncan3d 11602	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑋 + (𝑏 − 𝑋)) = 𝑏)
94	93	eqcomd 2731	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 = (𝑋 + (𝑏 − 𝑋)))
95	94	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → 𝑏 = (𝑋 + (𝑏 − 𝑋)))
96	95	opeq1d 4875	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + (𝑏 − 𝑋)), 𝑌⟩)
97	82, 87, 96	rspcedvd 3604	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → ∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
98	97	ex 411	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑦 = ⟨𝑏, 𝑌⟩ → ∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩))
99	98	rexlimdva 3145	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩ → ∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩))
100	78, 99	impbid 211	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩ ↔ ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩))
101	69, 100	bitrid 282	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩))
102		opeq2 4870	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑌 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑌⟩)
103	102	eqeq2d 2736	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑌 → (𝑦 = ⟨𝑏, 𝑐⟩ ↔ 𝑦 = ⟨𝑏, 𝑌⟩))
104	103	rexsng 4674	. . . . . . . . . . 11 ⊢ (𝑌 ∈ ℤ → (∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩ ↔ 𝑦 = ⟨𝑏, 𝑌⟩))
105	104	adantl 480	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩ ↔ 𝑦 = ⟨𝑏, 𝑌⟩))
106	105	bicomd 222	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑦 = ⟨𝑏, 𝑌⟩ ↔ ∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩))
107	106	rexbidv 3169	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩ ↔ ∃𝑏 ∈ ℤ ∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩))
108	101, 107	bitrd 278	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑏 ∈ ℤ ∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩))
109		eliun 4995	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩})
110		elxp2 5696	. . . . . . 7 ⊢ (𝑦 ∈ (ℤ × {𝑌}) ↔ ∃𝑏 ∈ ℤ ∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩)
111	108, 109, 110	3bitr4g 313	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑦 ∈ ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ 𝑦 ∈ (ℤ × {𝑌})))
112	111	eqrdv 2723	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} = (ℤ × {𝑌}))
113	62, 67, 112	3eqtrd 2769	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = (ℤ × {𝑌}))
114	52, 55, 113	3eqtrd 2769	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)} = (ℤ × {𝑌}))
115	22, 29, 114	3eqtrd 2769	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = (ℤ × {𝑌}))
116	18, 19, 115	3eqtrd 2769	1 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = (ℤ × {𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2702 ∃wrex 3060 ⊆ wss 3940 {csn 4624 ⟨cop 4630 ∪ ciun 4991 ↦ cmpt 5226 × cxp 5670 ran crn 5673 “ cima 5675 ‘cfv 6542 (class class class)co 7415 [cec 8719 ℂcc 11134 0cc0 11136 + caddc 11139 − cmin 11472 ℤcz 12586 Basecbs 17177 ↾_s cress 17206 +_gcplusg 17230 ×_s cxps 17485 Grpcgrp 18892 ~_QG cqg 19079 Rngcrng 20094 1_rcur 20123 Ringcrg 20175 ℤ_ringczring 21374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-addf 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-ec 8723 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-0g 17420 df-prds 17426 df-imas 17487 df-xps 17489 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-subg 19080 df-eqg 19082 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-subrng 20485 df-subrg 20510 df-cnfld 21282 df-zring 21375

This theorem is referenced by: pzriprnglem11 21419 pzriprnglem12 21420

W3C validator