jm2.27a - Mathbox for Stefan O'Rear

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Theorem jm2.27a 41729

Description: Lemma for jm2.27 41732. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)

**Hypotheses**
Ref	Expression
jm2.27a1	⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
jm2.27a2	⊢ (𝜑 → 𝐵 ∈ ℕ)
jm2.27a3	⊢ (𝜑 → 𝐶 ∈ ℕ)
jm2.27a4	⊢ (𝜑 → 𝐷 ∈ ℕ₀)
jm2.27a5	⊢ (𝜑 → 𝐸 ∈ ℕ₀)
jm2.27a6	⊢ (𝜑 → 𝐹 ∈ ℕ₀)
jm2.27a7	⊢ (𝜑 → 𝐺 ∈ ℕ₀)
jm2.27a8	⊢ (𝜑 → 𝐻 ∈ ℕ₀)
jm2.27a9	⊢ (𝜑 → 𝐼 ∈ ℕ₀)
jm2.27a10	⊢ (𝜑 → 𝐽 ∈ ℕ₀)
jm2.27a11	⊢ (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1)
jm2.27a12	⊢ (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1)
jm2.27a13	⊢ (𝜑 → 𝐺 ∈ (ℤ_≥‘2))
jm2.27a14	⊢ (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1)
jm2.27a15	⊢ (𝜑 → 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))))
jm2.27a16	⊢ (𝜑 → 𝐹 ∥ (𝐺 − 𝐴))
jm2.27a17	⊢ (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1))
jm2.27a18	⊢ (𝜑 → 𝐹 ∥ (𝐻 − 𝐶))
jm2.27a19	⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝐵))
jm2.27a20	⊢ (𝜑 → 𝐵 ≤ 𝐶)
jm2.27a21	⊢ (𝜑 → 𝑃 ∈ ℤ)
jm2.27a22	⊢ (𝜑 → 𝐷 = (𝐴 X_rm 𝑃))
jm2.27a23	⊢ (𝜑 → 𝐶 = (𝐴 Y_rm 𝑃))
jm2.27a24	⊢ (𝜑 → 𝑄 ∈ ℤ)
jm2.27a25	⊢ (𝜑 → 𝐹 = (𝐴 X_rm 𝑄))
jm2.27a26	⊢ (𝜑 → 𝐸 = (𝐴 Y_rm 𝑄))
jm2.27a27	⊢ (𝜑 → 𝑅 ∈ ℤ)
jm2.27a28	⊢ (𝜑 → 𝐼 = (𝐺 X_rm 𝑅))
jm2.27a29	⊢ (𝜑 → 𝐻 = (𝐺 Y_rm 𝑅))

**Assertion**
Ref	Expression
jm2.27a	⊢ (𝜑 → 𝐶 = (𝐴 Y_rm 𝐵))

**Proof of Theorem jm2.27a**
Step	Hyp	Ref	Expression
1		jm2.27a23	. 2 ⊢ (𝜑 → 𝐶 = (𝐴 Y_rm 𝑃))
2		2z 12590	. . . . . 6 ⊢ 2 ∈ ℤ
3		jm2.27a3	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℕ)
4	3	nnzd 12581	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ)
5		zmulcl 12607	. . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (2 · 𝐶) ∈ ℤ)
6	2, 4, 5	sylancr 587	. . . . 5 ⊢ (𝜑 → (2 · 𝐶) ∈ ℤ)
7		jm2.27a2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℕ)
8	7	nnzd 12581	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℤ)
9		jm2.27a27	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℤ)
10		jm2.27a21	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ)
11		jm2.27a8	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
12	11	nn0zd 12580	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ ℤ)
13		jm2.27a19	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝐵))
14		congsym 41692	. . . . . . . 8 ⊢ ((((2 · 𝐶) ∈ ℤ ∧ 𝐻 ∈ ℤ) ∧ (𝐵 ∈ ℤ ∧ (2 · 𝐶) ∥ (𝐻 − 𝐵))) → (2 · 𝐶) ∥ (𝐵 − 𝐻))
15	6, 12, 8, 13, 14	syl22anc 837	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐵 − 𝐻))
16		jm2.27a7	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ ℕ₀)
17	16	nn0zd 12580	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℤ)
18		peano2zm 12601	. . . . . . . . 9 ⊢ (𝐺 ∈ ℤ → (𝐺 − 1) ∈ ℤ)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐺 − 1) ∈ ℤ)
20	12, 9	zsubcld 12667	. . . . . . . 8 ⊢ (𝜑 → (𝐻 − 𝑅) ∈ ℤ)
21		jm2.27a17	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1))
22		jm2.27a13	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (ℤ_≥‘2))
23	11	nn0ge0d 12531	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ 𝐻)
24		rmy0 41653	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (ℤ_≥‘2) → (𝐺 Y_rm 0) = 0)
25	22, 24	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 Y_rm 0) = 0)
26		jm2.27a29	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐻 = (𝐺 Y_rm 𝑅))
27	26	eqcomd 2738	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 Y_rm 𝑅) = 𝐻)
28	23, 25, 27	3brtr4d 5179	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 Y_rm 0) ≤ (𝐺 Y_rm 𝑅))
29		0zd 12566	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℤ)
30		lermy 41679	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (ℤ_≥‘2) ∧ 0 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (0 ≤ 𝑅 ↔ (𝐺 Y_rm 0) ≤ (𝐺 Y_rm 𝑅)))
31	22, 29, 9, 30	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ 𝑅 ↔ (𝐺 Y_rm 0) ≤ (𝐺 Y_rm 𝑅)))
32	28, 31	mpbird 256	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝑅)
33		elnn0z 12567	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ₀ ↔ (𝑅 ∈ ℤ ∧ 0 ≤ 𝑅))
34	9, 32, 33	sylanbrc 583	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
35		jm2.16nn0 41728	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (ℤ_≥‘2) ∧ 𝑅 ∈ ℕ₀) → (𝐺 − 1) ∥ ((𝐺 Y_rm 𝑅) − 𝑅))
36	22, 34, 35	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 − 1) ∥ ((𝐺 Y_rm 𝑅) − 𝑅))
37	26	oveq1d 7420	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 − 𝑅) = ((𝐺 Y_rm 𝑅) − 𝑅))
38	36, 37	breqtrrd 5175	. . . . . . . 8 ⊢ (𝜑 → (𝐺 − 1) ∥ (𝐻 − 𝑅))
39	6, 19, 20, 21, 38	dvdstrd 16234	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝑅))
40		congtr 41689	. . . . . . 7 ⊢ ((((2 · 𝐶) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐻 ∈ ℤ ∧ 𝑅 ∈ ℤ) ∧ ((2 · 𝐶) ∥ (𝐵 − 𝐻) ∧ (2 · 𝐶) ∥ (𝐻 − 𝑅))) → (2 · 𝐶) ∥ (𝐵 − 𝑅))
41	6, 8, 12, 9, 15, 39, 40	syl222anc 1386	. . . . . 6 ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐵 − 𝑅))
42	41	orcd 871	. . . . 5 ⊢ (𝜑 → ((2 · 𝐶) ∥ (𝐵 − 𝑅) ∨ (2 · 𝐶) ∥ (𝐵 − -𝑅)))
43		jm2.27a24	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℤ)
44		zmulcl 12607	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (2 · 𝑄) ∈ ℤ)
45	2, 43, 44	sylancr 587	. . . . . 6 ⊢ (𝜑 → (2 · 𝑄) ∈ ℤ)
46		zsqcl 14090	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ ℤ → (𝐶↑2) ∈ ℤ)
47	4, 46	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶↑2) ∈ ℤ)
48		dvdsmul2 16218	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℤ ∧ (𝐶↑2) ∈ ℤ) → (𝐶↑2) ∥ (2 · (𝐶↑2)))
49	2, 47, 48	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶↑2) ∥ (2 · (𝐶↑2)))
50		jm2.27a10	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
51	50	nn0zd 12580	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ ℤ)
52	51	peano2zd 12665	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽 + 1) ∈ ℤ)
53		zmulcl 12607	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℤ ∧ (𝐶↑2) ∈ ℤ) → (2 · (𝐶↑2)) ∈ ℤ)
54	2, 47, 53	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (𝐶↑2)) ∈ ℤ)
55		dvdsmultr2 16237	. . . . . . . . . . . . 13 ⊢ (((𝐶↑2) ∈ ℤ ∧ (𝐽 + 1) ∈ ℤ ∧ (2 · (𝐶↑2)) ∈ ℤ) → ((𝐶↑2) ∥ (2 · (𝐶↑2)) → (𝐶↑2) ∥ ((𝐽 + 1) · (2 · (𝐶↑2)))))
56	47, 52, 54, 55	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶↑2) ∥ (2 · (𝐶↑2)) → (𝐶↑2) ∥ ((𝐽 + 1) · (2 · (𝐶↑2)))))
57	49, 56	mpd 15	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶↑2) ∥ ((𝐽 + 1) · (2 · (𝐶↑2))))
58	1	oveq1d 7420	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶↑2) = ((𝐴 Y_rm 𝑃)↑2))
59		jm2.27a15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))))
60		jm2.27a26	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 = (𝐴 Y_rm 𝑄))
61	59, 60	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐽 + 1) · (2 · (𝐶↑2))) = (𝐴 Y_rm 𝑄))
62	57, 58, 61	3brtr3d 5178	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 Y_rm 𝑃)↑2) ∥ (𝐴 Y_rm 𝑄))
63		jm2.27a1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
64	52	zred 12662	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 + 1) ∈ ℝ)
65	54	zred 12662	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 · (𝐶↑2)) ∈ ℝ)
66		nn0p1nn 12507	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ ℕ₀ → (𝐽 + 1) ∈ ℕ)
67	50, 66	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 + 1) ∈ ℕ)
68	67	nngt0d 12257	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < (𝐽 + 1))
69		2nn 12281	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℕ
70	3	nnsqcld 14203	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐶↑2) ∈ ℕ)
71		nnmulcl 12232	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℕ ∧ (𝐶↑2) ∈ ℕ) → (2 · (𝐶↑2)) ∈ ℕ)
72	69, 70, 71	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 · (𝐶↑2)) ∈ ℕ)
73	72	nngt0d 12257	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < (2 · (𝐶↑2)))
74	64, 65, 68, 73	mulgt0d 11365	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < ((𝐽 + 1) · (2 · (𝐶↑2))))
75	74, 59	breqtrrd 5175	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝐸)
76		rmy0 41653	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 Y_rm 0) = 0)
77	63, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 Y_rm 0) = 0)
78	60	eqcomd 2738	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 Y_rm 𝑄) = 𝐸)
79	75, 77, 78	3brtr4d 5179	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 Y_rm 0) < (𝐴 Y_rm 𝑄))
80		ltrmy 41676	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 0 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (0 < 𝑄 ↔ (𝐴 Y_rm 0) < (𝐴 Y_rm 𝑄)))
81	63, 29, 43, 80	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 𝑄 ↔ (𝐴 Y_rm 0) < (𝐴 Y_rm 𝑄)))
82	79, 81	mpbird 256	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑄)
83		elnnz 12564	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℕ ↔ (𝑄 ∈ ℤ ∧ 0 < 𝑄))
84	43, 82, 83	sylanbrc 583	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℕ)
85	3	nngt0d 12257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝐶)
86	1	eqcomd 2738	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 Y_rm 𝑃) = 𝐶)
87	85, 77, 86	3brtr4d 5179	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 Y_rm 0) < (𝐴 Y_rm 𝑃))
88		ltrmy 41676	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 0 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (0 < 𝑃 ↔ (𝐴 Y_rm 0) < (𝐴 Y_rm 𝑃)))
89	63, 29, 10, 88	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 𝑃 ↔ (𝐴 Y_rm 0) < (𝐴 Y_rm 𝑃)))
90	87, 89	mpbird 256	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑃)
91		elnnz 12564	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ ↔ (𝑃 ∈ ℤ ∧ 0 < 𝑃))
92	10, 90, 91	sylanbrc 583	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ)
93		jm2.20nn 41721	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑄 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (((𝐴 Y_rm 𝑃)↑2) ∥ (𝐴 Y_rm 𝑄) ↔ (𝑃 · (𝐴 Y_rm 𝑃)) ∥ 𝑄))
94	63, 84, 92, 93	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐴 Y_rm 𝑃)↑2) ∥ (𝐴 Y_rm 𝑄) ↔ (𝑃 · (𝐴 Y_rm 𝑃)) ∥ 𝑄))
95	62, 94	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 · (𝐴 Y_rm 𝑃)) ∥ 𝑄)
96	1, 4	eqeltrrd 2834	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 Y_rm 𝑃) ∈ ℤ)
97		muldvds2 16221	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ (𝐴 Y_rm 𝑃) ∈ ℤ ∧ 𝑄 ∈ ℤ) → ((𝑃 · (𝐴 Y_rm 𝑃)) ∥ 𝑄 → (𝐴 Y_rm 𝑃) ∥ 𝑄))
98	10, 96, 43, 97	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 · (𝐴 Y_rm 𝑃)) ∥ 𝑄 → (𝐴 Y_rm 𝑃) ∥ 𝑄))
99	95, 98	mpd 15	. . . . . . . 8 ⊢ (𝜑 → (𝐴 Y_rm 𝑃) ∥ 𝑄)
100	1, 99	eqbrtrd 5169	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∥ 𝑄)
101	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℤ)
102		dvdscmul 16222	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ 𝑄 ∈ ℤ ∧ 2 ∈ ℤ) → (𝐶 ∥ 𝑄 → (2 · 𝐶) ∥ (2 · 𝑄)))
103	4, 43, 101, 102	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∥ 𝑄 → (2 · 𝐶) ∥ (2 · 𝑄)))
104	100, 103	mpd 15	. . . . . 6 ⊢ (𝜑 → (2 · 𝐶) ∥ (2 · 𝑄))
105		jm2.27a25	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝐴 X_rm 𝑄))
106		jm2.27a6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ ℕ₀)
107	106	nn0zd 12580	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ℤ)
108	105, 107	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 X_rm 𝑄) ∈ ℤ)
109		frmy 41638	. . . . . . . . . . 11 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
110	109	fovcl 7533	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑅 ∈ ℤ) → (𝐴 Y_rm 𝑅) ∈ ℤ)
111	63, 9, 110	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 Y_rm 𝑅) ∈ ℤ)
112	26, 12	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 Y_rm 𝑅) ∈ ℤ)
113		eluzelz 12828	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℤ)
114	63, 113	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
115	114, 17	zsubcld 12667	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − 𝐺) ∈ ℤ)
116	111, 112	zsubcld 12667	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 Y_rm 𝑅) − (𝐺 Y_rm 𝑅)) ∈ ℤ)
117		jm2.27a16	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∥ (𝐺 − 𝐴))
118		congsym 41692	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℤ ∧ 𝐺 ∈ ℤ) ∧ (𝐴 ∈ ℤ ∧ 𝐹 ∥ (𝐺 − 𝐴))) → 𝐹 ∥ (𝐴 − 𝐺))
119	107, 17, 114, 117, 118	syl22anc 837	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∥ (𝐴 − 𝐺))
120	105, 119	eqbrtrrd 5171	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 X_rm 𝑄) ∥ (𝐴 − 𝐺))
121		jm2.15nn0 41727	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ (ℤ_≥‘2) ∧ 𝑅 ∈ ℕ₀) → (𝐴 − 𝐺) ∥ ((𝐴 Y_rm 𝑅) − (𝐺 Y_rm 𝑅)))
122	63, 22, 34, 121	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − 𝐺) ∥ ((𝐴 Y_rm 𝑅) − (𝐺 Y_rm 𝑅)))
123	108, 115, 116, 120, 122	dvdstrd 16234	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − (𝐺 Y_rm 𝑅)))
124		jm2.27a18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∥ (𝐻 − 𝐶))
125	26, 1	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝜑 → (𝐻 − 𝐶) = ((𝐺 Y_rm 𝑅) − (𝐴 Y_rm 𝑃)))
126	124, 105, 125	3brtr3d 5178	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 X_rm 𝑄) ∥ ((𝐺 Y_rm 𝑅) − (𝐴 Y_rm 𝑃)))
127		congtr 41689	. . . . . . . . 9 ⊢ ((((𝐴 X_rm 𝑄) ∈ ℤ ∧ (𝐴 Y_rm 𝑅) ∈ ℤ) ∧ ((𝐺 Y_rm 𝑅) ∈ ℤ ∧ (𝐴 Y_rm 𝑃) ∈ ℤ) ∧ ((𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − (𝐺 Y_rm 𝑅)) ∧ (𝐴 X_rm 𝑄) ∥ ((𝐺 Y_rm 𝑅) − (𝐴 Y_rm 𝑃)))) → (𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − (𝐴 Y_rm 𝑃)))
128	108, 111, 112, 96, 123, 126, 127	syl222anc 1386	. . . . . . . 8 ⊢ (𝜑 → (𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − (𝐴 Y_rm 𝑃)))
129	128	orcd 871	. . . . . . 7 ⊢ (𝜑 → ((𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − (𝐴 Y_rm 𝑃)) ∨ (𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − -(𝐴 Y_rm 𝑃))))
130		jm2.26 41726	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑄 ∈ ℕ) ∧ (𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ)) → (((𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − (𝐴 Y_rm 𝑃)) ∨ (𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − -(𝐴 Y_rm 𝑃))) ↔ ((2 · 𝑄) ∥ (𝑅 − 𝑃) ∨ (2 · 𝑄) ∥ (𝑅 − -𝑃))))
131	63, 84, 9, 10, 130	syl22anc 837	. . . . . . 7 ⊢ (𝜑 → (((𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − (𝐴 Y_rm 𝑃)) ∨ (𝐴 X_rm 𝑄) ∥ ((𝐴 Y_rm 𝑅) − -(𝐴 Y_rm 𝑃))) ↔ ((2 · 𝑄) ∥ (𝑅 − 𝑃) ∨ (2 · 𝑄) ∥ (𝑅 − -𝑃))))
132	129, 131	mpbid 231	. . . . . 6 ⊢ (𝜑 → ((2 · 𝑄) ∥ (𝑅 − 𝑃) ∨ (2 · 𝑄) ∥ (𝑅 − -𝑃)))
133		dvdsacongtr 41708	. . . . . 6 ⊢ ((((2 · 𝑄) ∈ ℤ ∧ 𝑅 ∈ ℤ) ∧ (𝑃 ∈ ℤ ∧ (2 · 𝐶) ∈ ℤ) ∧ ((2 · 𝐶) ∥ (2 · 𝑄) ∧ ((2 · 𝑄) ∥ (𝑅 − 𝑃) ∨ (2 · 𝑄) ∥ (𝑅 − -𝑃)))) → ((2 · 𝐶) ∥ (𝑅 − 𝑃) ∨ (2 · 𝐶) ∥ (𝑅 − -𝑃)))
134	45, 9, 10, 6, 104, 132, 133	syl222anc 1386	. . . . 5 ⊢ (𝜑 → ((2 · 𝐶) ∥ (𝑅 − 𝑃) ∨ (2 · 𝐶) ∥ (𝑅 − -𝑃)))
135		acongtr 41702	. . . . 5 ⊢ ((((2 · 𝐶) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ (((2 · 𝐶) ∥ (𝐵 − 𝑅) ∨ (2 · 𝐶) ∥ (𝐵 − -𝑅)) ∧ ((2 · 𝐶) ∥ (𝑅 − 𝑃) ∨ (2 · 𝐶) ∥ (𝑅 − -𝑃)))) → ((2 · 𝐶) ∥ (𝐵 − 𝑃) ∨ (2 · 𝐶) ∥ (𝐵 − -𝑃)))
136	6, 8, 9, 10, 42, 134, 135	syl222anc 1386	. . . 4 ⊢ (𝜑 → ((2 · 𝐶) ∥ (𝐵 − 𝑃) ∨ (2 · 𝐶) ∥ (𝐵 − -𝑃)))
137	7	nnnn0d 12528	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
138	3	nnnn0d 12528	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℕ₀)
139		jm2.27a20	. . . . . 6 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
140		elfz2nn0 13588	. . . . . 6 ⊢ (𝐵 ∈ (0...𝐶) ↔ (𝐵 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀ ∧ 𝐵 ≤ 𝐶))
141	137, 138, 139, 140	syl3anbrc 1343	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (0...𝐶))
142	92	nnnn0d 12528	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
143		rmygeid 41688	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℕ₀) → 𝑃 ≤ (𝐴 Y_rm 𝑃))
144	63, 142, 143	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≤ (𝐴 Y_rm 𝑃))
145	144, 1	breqtrrd 5175	. . . . . 6 ⊢ (𝜑 → 𝑃 ≤ 𝐶)
146		elfz2nn0 13588	. . . . . 6 ⊢ (𝑃 ∈ (0...𝐶) ↔ (𝑃 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀ ∧ 𝑃 ≤ 𝐶))
147	142, 138, 145, 146	syl3anbrc 1343	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (0...𝐶))
148		acongeq 41707	. . . . 5 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ (0...𝐶) ∧ 𝑃 ∈ (0...𝐶)) → (𝐵 = 𝑃 ↔ ((2 · 𝐶) ∥ (𝐵 − 𝑃) ∨ (2 · 𝐶) ∥ (𝐵 − -𝑃))))
149	3, 141, 147, 148	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐵 = 𝑃 ↔ ((2 · 𝐶) ∥ (𝐵 − 𝑃) ∨ (2 · 𝐶) ∥ (𝐵 − -𝑃))))
150	136, 149	mpbird 256	. . 3 ⊢ (𝜑 → 𝐵 = 𝑃)
151	150	oveq2d 7421	. 2 ⊢ (𝜑 → (𝐴 Y_rm 𝐵) = (𝐴 Y_rm 𝑃))
152	1, 151	eqtr4d 2775	1 ⊢ (𝜑 → 𝐶 = (𝐴 Y_rm 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 -cneg 11441 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ↑cexp 14023 ∥ cdvds 16193 X_rm crmx 41623 Y_rm crmy 41624

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-dvds 16194 df-gcd 16432 df-prm 16605 df-numer 16667 df-denom 16668 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-squarenn 41564 df-pell1qr 41565 df-pell14qr 41566 df-pell1234qr 41567 df-pellfund 41568 df-rmx 41625 df-rmy 41626

This theorem is referenced by: jm2.27b 41730

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