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Theorem pzriprnglem10 21501

Description: Lemma 10 for pzriprng 21508: 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 21492	. . . 4 ⊢ 𝑅 ∈ Rng
3		rnggrp 20155	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
4	2, 3	ax-mp 5	. . 3 ⊢ 𝑅 ∈ Grp
5		pzriprng.i	. . . . 5 ⊢ 𝐼 = (ℤ × {0})
6		0z 12624	. . . . . 6 ⊢ 0 ∈ ℤ
7		snssi 4808	. . . . . 6 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
8		xpss2 5705	. . . . . 6 ⊢ ({0} ⊆ ℤ → (ℤ × {0}) ⊆ (ℤ × ℤ))
9	6, 7, 8	mp2b 10	. . . . 5 ⊢ (ℤ × {0}) ⊆ (ℤ × ℤ)
10	5, 9	eqsstri 4030	. . . 4 ⊢ 𝐼 ⊆ (ℤ × ℤ)
11	10	a1i 11	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝐼 ⊆ (ℤ × ℤ))
12		opelxpi 5722	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ⟨𝑋, 𝑌⟩ ∈ (ℤ × ℤ))
13	1	pzriprnglem2 21493	. . . . 5 ⊢ (Base‘𝑅) = (ℤ × ℤ)
14	13	eqcomi 2746	. . . 4 ⊢ (ℤ × ℤ) = (Base‘𝑅)
15		pzriprng.g	. . . 4 ⊢ ∼ = (𝑅 ~_QG 𝐼)
16		eqid 2737	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
17	14, 15, 16	eqglact 19197	. . 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 6091	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((𝑥 ∈ (ℤ × ℤ) ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) “ 𝐼) = ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)))
20		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥))
21	20	rnmpt 5968	. . . 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 3325	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ ∃𝑥 ∈ (ℤ × {0})𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥))
24		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩))
25	24	eqeq2d 2748	. . . . . . 7 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)))
26	25	rexxp 5853	. . . . . 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 2808	. . 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 11255	. . . . . . . 8 ⊢ 0 ∈ V
31		opeq2 4874	. . . . . . . . . 10 ⊢ (𝑏 = 0 → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 0⟩)
32	31	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑏 = 0 → (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩) = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩))
33	32	eqeq2d 2748	. . . . . . . 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 21464	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤ_ring)
36		zringring 21460	. . . . . . . . . 10 ⊢ ℤ_ring ∈ Ring
37	36	a1i 11	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → ℤ_ring ∈ Ring)
38		simpll 767	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 𝑋 ∈ ℤ)
39		simplr 769	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 𝑌 ∈ ℤ)
40		simpr 484	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 𝑎 ∈ ℤ)
41		0zd 12625	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 0 ∈ ℤ)
42	38, 40	zaddcld 12726	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑋 + 𝑎) ∈ ℤ)
43		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑌 ∈ ℤ)
44		0zd 12625	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 0 ∈ ℤ)
45	43, 44	zaddcld 12726	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 + 0) ∈ ℤ)
46	45	adantr 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑌 + 0) ∈ ℤ)
47		zringplusg 21465	. . . . . . . . 9 ⊢ + = (+_g‘ℤ_ring)
48	1, 35, 35, 37, 37, 38, 39, 40, 41, 42, 46, 47, 47, 16	xpsadd 17619	. . . . . . . 8 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 0⟩) = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩)
49	48	eqeq2d 2748	. . . . . . 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 3177	. . . . 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 2808	. . . 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 5051	. . . . . 6 ⊢ ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = {𝑒 ∣ ∃𝑎 ∈ ℤ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩}
54	53	eqcomi 2746	. . . . 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 12618	. . . . . . . . . . 11 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℂ)
57	56	adantl 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑌 ∈ ℂ)
58	57	addridd 11461	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 + 0) = 𝑌)
59	58	opeq2d 4880	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
60	59	eqeq2d 2748	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩ ↔ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩))
61	60	abbidv 2808	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩})
62	61	iuneq2d 5022	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩})
63		df-sn 4627	. . . . . . . . 9 ⊢ {⟨(𝑋 + 𝑎), 𝑌⟩} = {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩}
64	63	eqcomi 2746	. . . . . . . 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 5013	. . . . . 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 4642	. . . . . . . . . 10 ⊢ (𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
69	68	rexbii 3094	. . . . . . . . 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 769	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) ∧ 𝑏 = (𝑋 + 𝑎)) → 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩)
72		opeq1 4873	. . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) ∧ 𝑦 = ⟨(𝑋 + 𝑎), 𝑌⟩) → ⟨(𝑋 + 𝑎), 𝑌⟩ = ⟨(𝑋 + 𝑎), 𝑌⟩)
76	70, 74, 75	rspcedvd 3624	. . . . . . . . . . . 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 3155	. . . . . . . . . 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 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑋 ∈ ℤ)
81	79, 80	zsubcld 12727	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → 𝑦 = ⟨𝑏, 𝑌⟩)
84		oveq2 7439	. . . . . . . . . . . . . . . 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 4879	. . . . . . . . . . . . . 14 ⊢ (((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) ∧ 𝑎 = (𝑏 − 𝑋)) → ⟨(𝑋 + 𝑎), 𝑌⟩ = ⟨(𝑋 + (𝑏 − 𝑋)), 𝑌⟩)
87	83, 86	eqeq12d 2753	. . . . . . . . . . . . 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 12618	. . . . . . . . . . . . . . . . . . 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 12618	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ)
92	91	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
93	90, 92	pncan3d 11623	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → (𝑋 + (𝑏 − 𝑋)) = 𝑏)
94	93	eqcomd 2743	. . . . . . . . . . . . . . 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 4879	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + (𝑏 − 𝑋)), 𝑌⟩)
97	82, 87, 96	rspcedvd 3624	. . . . . . . . . . . 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 3155	. . . . . . . . . 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 4874	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑌 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑌⟩)
103	102	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑌 → (𝑦 = ⟨𝑏, 𝑐⟩ ↔ 𝑦 = ⟨𝑏, 𝑌⟩))
104	103	rexsng 4676	. . . . . . . . . . 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 3179	. . . . . . . 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 4995	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩})
110		elxp2 5709	. . . . . . 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 2735	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} = (ℤ × {𝑌}))
113	62, 67, 112	3eqtrd 2781	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = (ℤ × {𝑌}))
114	52, 55, 113	3eqtrd 2781	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)} = (ℤ × {𝑌}))
115	22, 29, 114	3eqtrd 2781	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = (ℤ × {𝑌}))
116	18, 19, 115	3eqtrd 2781	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 2714 ∃wrex 3070 ⊆ wss 3951 {csn 4626 ⟨cop 4632 ∪ ciun 4991 ↦ cmpt 5225 × cxp 5683 ran crn 5686 “ cima 5688 ‘cfv 6561 (class class class)co 7431 [cec 8743 ℂcc 11153 0cc0 11155 + caddc 11158 − cmin 11492 ℤcz 12613 Basecbs 17247 ↾_s cress 17274 +_gcplusg 17297 ×_s cxps 17551 Grpcgrp 18951 ~_QG cqg 19140 Rngcrng 20149 1_rcur 20178 Ringcrg 20230 ℤ_ringczring 21457

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17486 df-prds 17492 df-imas 17553 df-xps 17555 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-eqg 19143 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-cnfld 21365 df-zring 21458

This theorem is referenced by: pzriprnglem11 21502 pzriprnglem12 21503

W3C validator