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Theorem pzriprnglem10 21449

Description: Lemma 10 for pzriprng 21456: 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 21440	. . . 4 ⊢ 𝑅 ∈ Rng
3		rnggrp 20116	. . . 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 4784	. . . . . 6 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
8		xpss2 5674	. . . . . 6 ⊢ ({0} ⊆ ℤ → (ℤ × {0}) ⊆ (ℤ × ℤ))
9	6, 7, 8	mp2b 10	. . . . 5 ⊢ (ℤ × {0}) ⊆ (ℤ × ℤ)
10	5, 9	eqsstri 4005	. . . 4 ⊢ 𝐼 ⊆ (ℤ × ℤ)
11	10	a1i 11	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝐼 ⊆ (ℤ × ℤ))
12		opelxpi 5691	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ⟨𝑋, 𝑌⟩ ∈ (ℤ × ℤ))
13	1	pzriprnglem2 21441	. . . . 5 ⊢ (Base‘𝑅) = (ℤ × ℤ)
14	13	eqcomi 2744	. . . 4 ⊢ (ℤ × ℤ) = (Base‘𝑅)
15		pzriprng.g	. . . 4 ⊢ ∼ = (𝑅 ~_QG 𝐼)
16		eqid 2735	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
17	14, 15, 16	eqglact 19160	. . 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 1468	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = ((𝑥 ∈ (ℤ × ℤ) ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) “ 𝐼))
19	11	mptimass 6060	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((𝑥 ∈ (ℤ × ℤ) ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) “ 𝐼) = ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)))
20		eqid 2735	. . . . 5 ⊢ (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥))
21	20	rnmpt 5937	. . . 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 3304	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ ∃𝑥 ∈ (ℤ × {0})𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥))
24		oveq2 7411	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩))
25	24	eqeq2d 2746	. . . . . . 7 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)))
26	25	rexxp 5822	. . . . . 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 275	. . . . 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 2801	. . 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 11227	. . . . . . . 8 ⊢ 0 ∈ V
31		opeq2 4850	. . . . . . . . . 10 ⊢ (𝑏 = 0 → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 0⟩)
32	31	oveq2d 7419	. . . . . . . . 9 ⊢ (𝑏 = 0 → (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩))
33	32	eqeq2d 2746	. . . . . . . 8 ⊢ (𝑏 = 0 → (𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) ↔ 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩)))
34	30, 33	rexsn 4658	. . . . . . 7 ⊢ (∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) ↔ 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩))
35		zringbas 21412	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤ_ring)
36		zringring 21408	. . . . . . . . . 10 ⊢ ℤ_ring ∈ Ring
37	36	a1i 11	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → ℤ_ring ∈ Ring)
38		simpll 766	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 𝑋 ∈ ℤ)
39		simplr 768	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 𝑌 ∈ ℤ)
40		simpr 484	. . . . . . . . 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 12699	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑋 + 𝑎) ∈ ℤ)
43		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑌 ∈ ℤ)
44		0zd 12598	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 0 ∈ ℤ)
45	43, 44	zaddcld 12699	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 + 0) ∈ ℤ)
46	45	adantr 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑌 + 0) ∈ ℤ)
47		zringplusg 21413	. . . . . . . . 9 ⊢ + = (+_g‘ℤ_ring)
48	1, 35, 35, 37, 37, 38, 39, 40, 41, 42, 46, 47, 47, 16	xpsadd 17586	. . . . . . . 8 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩) = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩)
49	48	eqeq2d 2746	. . . . . . 7 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩) ↔ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩))
50	34, 49	bitrid 283	. . . . . 6 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) ↔ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩))
51	50	rexbidva 3162	. . . . 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 2801	. . . 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 5027	. . . . . 6 ⊢ ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = {𝑒 ∣ ∃𝑎 ∈ ℤ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩}
54	53	eqcomi 2744	. . . . 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 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑌 ∈ ℂ)
58	57	addridd 11433	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 + 0) = 𝑌)
59	58	opeq2d 4856	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
60	59	eqeq2d 2746	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩ ↔ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩))
61	60	abbidv 2801	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩})
62	61	iuneq2d 4998	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩})
63		df-sn 4602	. . . . . . . . 9 ⊢ {⟨(𝑋 + 𝑎), 𝑌⟩} = {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩}
64	63	eqcomi 2744	. . . . . . . 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 4989	. . . . . 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 4617	. . . . . . . . . 10 ⊢ (𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
69	68	rexbii 3083	. . . . . . . . 9 ⊢ (∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
70	42	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) → (𝑋 + 𝑎) ∈ ℤ)
71		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) ∧ 𝑏 = (𝑋 + 𝑎)) → 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
72		opeq1 4849	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑋 + 𝑎) → ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
73	72	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) ∧ 𝑏 = (𝑋 + 𝑎)) → ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
74	71, 73	eqeq12d 2751	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) → ⟨(𝑋 + 𝑎), 𝑌⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
76	70, 74, 75	rspcedvd 3603	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) → ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩)
77	76	ex 412	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩ → ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩))
78	77	rexlimdva 3141	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩ → ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩))
79		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℤ)
80		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑋 ∈ ℤ)
81	79, 80	zsubcld 12700	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑏 − 𝑋) ∈ ℤ)
82	81	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → (𝑏 − 𝑋) ∈ ℤ)
83		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → 𝑦 = ⟨𝑏, 𝑌⟩)
84		oveq2 7411	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑏 − 𝑋) → (𝑋 + 𝑎) = (𝑋 + (𝑏 − 𝑋)))
85	84	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → (𝑋 + 𝑎) = (𝑋 + (𝑏 − 𝑋)))
86	85	opeq1d 4855	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → ⟨(𝑋 + 𝑎), 𝑌⟩ = ⟨(𝑋 + (𝑏 − 𝑋)), 𝑌⟩)
87	83, 86	eqeq12d 2751	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑋 ∈ ℂ)
90	89	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑋 ∈ ℂ)
91		zcn 12591	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ)
92	91	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
93	90, 92	pncan3d 11595	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑋 + (𝑏 − 𝑋)) = 𝑏)
94	93	eqcomd 2741	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 = (𝑋 + (𝑏 − 𝑋)))
95	94	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → 𝑏 = (𝑋 + (𝑏 − 𝑋)))
96	95	opeq1d 4855	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + (𝑏 − 𝑋)), 𝑌⟩)
97	82, 87, 96	rspcedvd 3603	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → ∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
98	97	ex 412	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑦 = ⟨𝑏, 𝑌⟩ → ∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩))
99	98	rexlimdva 3141	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩ → ∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩))
100	78, 99	impbid 212	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩ ↔ ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩))
101	69, 100	bitrid 283	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩))
102		opeq2 4850	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑌 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑌⟩)
103	102	eqeq2d 2746	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑌 → (𝑦 = ⟨𝑏, 𝑐⟩ ↔ 𝑦 = ⟨𝑏, 𝑌⟩))
104	103	rexsng 4652	. . . . . . . . . . 11 ⊢ (𝑌 ∈ ℤ → (∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩ ↔ 𝑦 = ⟨𝑏, 𝑌⟩))
105	104	adantl 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩ ↔ 𝑦 = ⟨𝑏, 𝑌⟩))
106	105	bicomd 223	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑦 = ⟨𝑏, 𝑌⟩ ↔ ∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩))
107	106	rexbidv 3164	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑏 ∈ ℤ 𝑦 = ⟨𝑏, 𝑌⟩ ↔ ∃𝑏 ∈ ℤ ∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩))
108	101, 107	bitrd 279	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑏 ∈ ℤ ∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩))
109		eliun 4971	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩})
110		elxp2 5678	. . . . . . 7 ⊢ (𝑦 ∈ (ℤ × {𝑌}) ↔ ∃𝑏 ∈ ℤ ∃𝑐 ∈ {𝑌}𝑦 = ⟨𝑏, 𝑐⟩)
111	108, 109, 110	3bitr4g 314	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑦 ∈ ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ 𝑦 ∈ (ℤ × {𝑌})))
112	111	eqrdv 2733	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} = (ℤ × {𝑌}))
113	62, 67, 112	3eqtrd 2774	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = (ℤ × {𝑌}))
114	52, 55, 113	3eqtrd 2774	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)} = (ℤ × {𝑌}))
115	22, 29, 114	3eqtrd 2774	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = (ℤ × {𝑌}))
116	18, 19, 115	3eqtrd 2774	1 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = (ℤ × {𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2713 ∃wrex 3060 ⊆ wss 3926 {csn 4601 ⟨cop 4607 ∪ ciun 4967 ↦ cmpt 5201 × cxp 5652 ran crn 5655 “ cima 5657 ‘cfv 6530 (class class class)co 7403 [cec 8715 ℂcc 11125 0cc0 11127 + caddc 11130 − cmin 11464 ℤcz 12586 Basecbs 17226 ↾_s cress 17249 +_gcplusg 17269 ×_s cxps 17518 Grpcgrp 18914 ~_QG cqg 19103 Rngcrng 20110 1_rcur 20139 Ringcrg 20191 ℤ_ringczring 21405

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-addf 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-ec 8719 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9452 df-inf 9453 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-fz 13523 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-0g 17453 df-prds 17459 df-imas 17520 df-xps 17522 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-grp 18917 df-minusg 18918 df-subg 19104 df-eqg 19106 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-cring 20194 df-subrng 20504 df-subrg 20528 df-cnfld 21314 df-zring 21406

This theorem is referenced by: pzriprnglem11 21450 pzriprnglem12 21451

W3C validator