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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 20097	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
4	2, 3	ax-mp 5	. . 3 ⊢ 𝑅 ∈ Grp
5		pzriprng.i	. . . . 5 ⊢ 𝐼 = (ℤ × {0})
6		0z 12503	. . . . . 6 ⊢ 0 ∈ ℤ
7		snssi 4765	. . . . . 6 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
8		xpss2 5645	. . . . . 6 ⊢ ({0} ⊆ ℤ → (ℤ × {0}) ⊆ (ℤ × ℤ))
9	6, 7, 8	mp2b 10	. . . . 5 ⊢ (ℤ × {0}) ⊆ (ℤ × ℤ)
10	5, 9	eqsstri 3981	. . . 4 ⊢ 𝐼 ⊆ (ℤ × ℤ)
11	10	a1i 11	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝐼 ⊆ (ℤ × ℤ))
12		opelxpi 5662	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ⟨𝑋, 𝑌⟩ ∈ (ℤ × ℤ))
13	1	pzriprnglem2 21441	. . . . 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 19112	. . 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 1469	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = ((𝑥 ∈ (ℤ × ℤ) ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) “ 𝐼))
19	11	mptimass 6033	. 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 5907	. . . 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 3296	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐼 𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥) ↔ ∃𝑥 ∈ (ℤ × {0})𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥))
24		oveq2 7368	. . . . . . . 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 5792	. . . . . 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 2803	. . 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 11130	. . . . . . . 8 ⊢ 0 ∈ V
31		opeq2 4831	. . . . . . . . . 10 ⊢ (𝑏 = 0 → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 0⟩)
32	31	oveq2d 7376	. . . . . . . . 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 4640	. . . . . . 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 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 12504	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → 0 ∈ ℤ)
42	38, 40	zaddcld 12604	. . . . . . . . 9 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → (𝑋 + 𝑎) ∈ ℤ)
43		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑌 ∈ ℤ)
44		0zd 12504	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 0 ∈ ℤ)
45	43, 44	zaddcld 12604	. . . . . . . . . 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 17499	. . . . . . . 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 3159	. . . . 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 2803	. . . 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 5008	. . . . . 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 12497	. . . . . . . . . . 11 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℂ)
57	56	adantl 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → 𝑌 ∈ ℂ)
58	57	addridd 11337	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 + 0) = 𝑌)
59	58	opeq2d 4837	. . . . . . . 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 2803	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩})
62	61	iuneq2d 4978	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), 𝑌⟩})
63		df-sn 4582	. . . . . . . . 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 4969	. . . . . 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 4597	. . . . . . . . . 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 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 4830	. . . . . . . . . . . . . . 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 3579	. . . . . . . . . . . 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 3138	. . . . . . . . . 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 12605	. . . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . . 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 4836	. . . . . . . . . . . . . 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 12497	. . . . . . . . . . . . . . . . . . 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 12497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ)
92	91	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
93	90, 92	pncan3d 11499	. . . . . . . . . . . . . . . 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 4836	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ 𝑏 ∈ ℤ) ∧ 𝑦 = ⟨𝑏, 𝑌⟩) → ⟨𝑏, 𝑌⟩ = ⟨(𝑋 + (𝑏 − 𝑋)), 𝑌⟩)
97	82, 87, 96	rspcedvd 3579	. . . . . . . . . . . 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 3138	. . . . . . . . . 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 4831	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑌 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑌⟩)
103	102	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑌 → (𝑦 = ⟨𝑏, 𝑐⟩ ↔ 𝑦 = ⟨𝑏, 𝑌⟩))
104	103	rexsng 4634	. . . . . . . . . . 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 3161	. . . . . . . 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 4951	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑎 ∈ ℤ {⟨(𝑋 + 𝑎), 𝑌⟩} ↔ ∃𝑎 ∈ ℤ 𝑦 ∈ {⟨(𝑋 + 𝑎), 𝑌⟩})
110		elxp2 5649	. . . . . . 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 2776	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ∪ 𝑎 ∈ ℤ {𝑒 ∣ 𝑒 = ⟨(𝑋 + 𝑎), (𝑌 + 0)⟩} = (ℤ × {𝑌}))
114	52, 55, 113	3eqtrd 2776	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → {𝑒 ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ {0}𝑒 = (⟨𝑋, 𝑌⟩(+_g‘𝑅)⟨𝑎, 𝑏⟩)} = (ℤ × {𝑌}))
115	22, 29, 114	3eqtrd 2776	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ran (𝑥 ∈ 𝐼 ↦ (⟨𝑋, 𝑌⟩(+_g‘𝑅)𝑥)) = (ℤ × {𝑌}))
116	18, 19, 115	3eqtrd 2776	1 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = (ℤ × {𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {cab 2715 ∃wrex 3061 ⊆ wss 3902 {csn 4581 ⟨cop 4587 ∪ ciun 4947 ↦ cmpt 5180 × cxp 5623 ran crn 5626 “ cima 5628 ‘cfv 6493 (class class class)co 7360 [cec 8635 ℂcc 11028 0cc0 11030 + caddc 11033 − cmin 11368 ℤcz 12492 Basecbs 17140 ↾_s cress 17161 +_gcplusg 17181 ×_s cxps 17431 Grpcgrp 18867 ~_QG cqg 19056 Rngcrng 20091 1_rcur 20120 Ringcrg 20172 ℤ_ringczring 21405

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-addf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-ec 8639 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-fz 13428 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-0g 17365 df-prds 17371 df-imas 17433 df-xps 17435 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18870 df-minusg 18871 df-subg 19057 df-eqg 19059 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-subrng 20483 df-subrg 20507 df-cnfld 21314 df-zring 21406

This theorem is referenced by: pzriprnglem11 21450 pzriprnglem12 21451

W3C validator