jm2.26a - Mathbox for Stefan O'Rear

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Theorem jm2.26a 41739

Description: Lemma for jm2.26 41741. Reverse direction is required to prove forward direction, so do it separately. Induction on difference between K and M, together with the addition formula fact that adding 2N only inverts sign. (Contributed by Stefan O'Rear, 2-Oct-2014.)

**Assertion**
Ref	Expression
jm2.26a	⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((2 · 𝑁) ∥ (𝐾 − 𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀)) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)))))

Proof of Theorem jm2.26a Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2z 12594	. . . . 5 ⊢ 2 ∈ ℤ
2		simplr 768	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℤ)
3		zmulcl 12611	. . . . 5 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 · 𝑁) ∈ ℤ)
4	1, 2, 3	sylancr 588	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (2 · 𝑁) ∈ ℤ)
5		zsubcl 12604	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 − 𝑀) ∈ ℤ)
6	5	adantl 483	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝐾 − 𝑀) ∈ ℤ)
7		divides 16199	. . . 4 ⊢ (((2 · 𝑁) ∈ ℤ ∧ (𝐾 − 𝑀) ∈ ℤ) → ((2 · 𝑁) ∥ (𝐾 − 𝑀) ↔ ∃𝑎 ∈ ℤ (𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀)))
8	4, 6, 7	syl2anc 585	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((2 · 𝑁) ∥ (𝐾 − 𝑀) ↔ ∃𝑎 ∈ ℤ (𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀)))
9		simplll 774	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → 𝐴 ∈ (ℤ_≥‘2))
10		simplrr 777	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → 𝑀 ∈ ℤ)
11		simpllr 775	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → 𝑁 ∈ ℤ)
12		simpr 486	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → 𝑎 ∈ ℤ)
13		jm2.25 41738	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) − -(𝐴 Y_rm 𝑀))))
14	9, 10, 11, 12, 13	syl121anc 1376	. . . . . 6 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) − -(𝐴 Y_rm 𝑀))))
15	14	adantr 482	. . . . 5 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀)) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) − -(𝐴 Y_rm 𝑀))))
16		oveq2 7417	. . . . . . . 8 ⊢ ((𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀) → (𝑀 + (𝑎 · (2 · 𝑁))) = (𝑀 + (𝐾 − 𝑀)))
17	16	oveq2d 7425	. . . . . . 7 ⊢ ((𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀) → (𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) = (𝐴 Y_rm (𝑀 + (𝐾 − 𝑀))))
18		zcn 12563	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
19		zcn 12563	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
20		pncan3 11468	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (𝑀 + (𝐾 − 𝑀)) = 𝐾)
21	18, 19, 20	syl2anr 598	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 + (𝐾 − 𝑀)) = 𝐾)
22	21	ad2antlr 726	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → (𝑀 + (𝐾 − 𝑀)) = 𝐾)
23	22	oveq2d 7425	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → (𝐴 Y_rm (𝑀 + (𝐾 − 𝑀))) = (𝐴 Y_rm 𝐾))
24	17, 23	sylan9eqr 2795	. . . . . 6 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀)) → (𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) = (𝐴 Y_rm 𝐾))
25		eqidd 2734	. . . . . 6 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀)) → (𝐴 Y_rm 𝑀) = (𝐴 Y_rm 𝑀))
26	24, 25	acongeq12d 41718	. . . . 5 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀)) → (((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (𝑀 + (𝑎 · (2 · 𝑁)))) − -(𝐴 Y_rm 𝑀))) ↔ ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)))))
27	15, 26	mpbid 231	. . . 4 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀)) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀))))
28	27	rexlimdva2 3158	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (∃𝑎 ∈ ℤ (𝑎 · (2 · 𝑁)) = (𝐾 − 𝑀) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)))))
29	8, 28	sylbid 239	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((2 · 𝑁) ∥ (𝐾 − 𝑀) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)))))
30		simprl 770	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐾 ∈ ℤ)
31		znegcl 12597	. . . . . 6 ⊢ (𝑀 ∈ ℤ → -𝑀 ∈ ℤ)
32	31	ad2antll 728	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → -𝑀 ∈ ℤ)
33	30, 32	zsubcld 12671	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝐾 − -𝑀) ∈ ℤ)
34		divides 16199	. . . 4 ⊢ (((2 · 𝑁) ∈ ℤ ∧ (𝐾 − -𝑀) ∈ ℤ) → ((2 · 𝑁) ∥ (𝐾 − -𝑀) ↔ ∃𝑎 ∈ ℤ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)))
35	4, 33, 34	syl2anc 585	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((2 · 𝑁) ∥ (𝐾 − -𝑀) ↔ ∃𝑎 ∈ ℤ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)))
36		frmx 41652	. . . . . . . . . 10 ⊢ X_rm :((ℤ_≥‘2) × ℤ)⟶ℕ₀
37	36	fovcl 7537	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 X_rm 𝑁) ∈ ℕ₀)
38	37	nn0zd 12584	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 X_rm 𝑁) ∈ ℤ)
39	9, 11, 38	syl2anc 585	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → (𝐴 X_rm 𝑁) ∈ ℤ)
40		simplrl 776	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → 𝐾 ∈ ℤ)
41		frmy 41653	. . . . . . . . 9 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
42	41	fovcl 7537	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝐾 ∈ ℤ) → (𝐴 Y_rm 𝐾) ∈ ℤ)
43	9, 40, 42	syl2anc 585	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → (𝐴 Y_rm 𝐾) ∈ ℤ)
44	41	fovcl 7537	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∈ ℤ)
45	9, 10, 44	syl2anc 585	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∈ ℤ)
46	39, 43, 45	3jca 1129	. . . . . 6 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → ((𝐴 X_rm 𝑁) ∈ ℤ ∧ (𝐴 Y_rm 𝐾) ∈ ℤ ∧ (𝐴 Y_rm 𝑀) ∈ ℤ))
47	46	adantr 482	. . . . 5 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)) → ((𝐴 X_rm 𝑁) ∈ ℤ ∧ (𝐴 Y_rm 𝐾) ∈ ℤ ∧ (𝐴 Y_rm 𝑀) ∈ ℤ))
48	32	adantr 482	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → -𝑀 ∈ ℤ)
49		jm2.25 41738	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (-𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑎 ∈ ℤ) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) − (𝐴 Y_rm -𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) − -(𝐴 Y_rm -𝑀))))
50	9, 48, 11, 12, 49	syl121anc 1376	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) − (𝐴 Y_rm -𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) − -(𝐴 Y_rm -𝑀))))
51	50	adantr 482	. . . . . 6 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) − (𝐴 Y_rm -𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) − -(𝐴 Y_rm -𝑀))))
52		oveq2 7417	. . . . . . . . 9 ⊢ ((𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀) → (-𝑀 + (𝑎 · (2 · 𝑁))) = (-𝑀 + (𝐾 − -𝑀)))
53	52	oveq2d 7425	. . . . . . . 8 ⊢ ((𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀) → (𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) = (𝐴 Y_rm (-𝑀 + (𝐾 − -𝑀))))
54	18	negcld 11558	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → -𝑀 ∈ ℂ)
55		pncan3 11468	. . . . . . . . . . 11 ⊢ ((-𝑀 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (-𝑀 + (𝐾 − -𝑀)) = 𝐾)
56	54, 19, 55	syl2anr 598	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (-𝑀 + (𝐾 − -𝑀)) = 𝐾)
57	56	ad2antlr 726	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → (-𝑀 + (𝐾 − -𝑀)) = 𝐾)
58	57	oveq2d 7425	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → (𝐴 Y_rm (-𝑀 + (𝐾 − -𝑀))) = (𝐴 Y_rm 𝐾))
59	53, 58	sylan9eqr 2795	. . . . . . 7 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)) → (𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) = (𝐴 Y_rm 𝐾))
60		rmyneg 41667	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Y_rm -𝑀) = -(𝐴 Y_rm 𝑀))
61	9, 10, 60	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) → (𝐴 Y_rm -𝑀) = -(𝐴 Y_rm 𝑀))
62	61	adantr 482	. . . . . . 7 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)) → (𝐴 Y_rm -𝑀) = -(𝐴 Y_rm 𝑀))
63	59, 62	acongeq12d 41718	. . . . . 6 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)) → (((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) − (𝐴 Y_rm -𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm (-𝑀 + (𝑎 · (2 · 𝑁)))) − -(𝐴 Y_rm -𝑀))) ↔ ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − --(𝐴 Y_rm 𝑀)))))
64	51, 63	mpbid 231	. . . . 5 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − --(𝐴 Y_rm 𝑀))))
65		acongneg2 41716	. . . . 5 ⊢ ((((𝐴 X_rm 𝑁) ∈ ℤ ∧ (𝐴 Y_rm 𝐾) ∈ ℤ ∧ (𝐴 Y_rm 𝑀) ∈ ℤ) ∧ ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − --(𝐴 Y_rm 𝑀)))) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀))))
66	47, 64, 65	syl2anc 585	. . . 4 ⊢ (((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀)) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀))))
67	66	rexlimdva2 3158	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (∃𝑎 ∈ ℤ (𝑎 · (2 · 𝑁)) = (𝐾 − -𝑀) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)))))
68	35, 67	sylbid 239	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((2 · 𝑁) ∥ (𝐾 − -𝑀) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)))))
69	29, 68	jaod 858	1 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((2 · 𝑁) ∥ (𝐾 − 𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀)) → ((𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − (𝐴 Y_rm 𝑀)) ∨ (𝐴 X_rm 𝑁) ∥ ((𝐴 Y_rm 𝐾) − -(𝐴 Y_rm 𝑀)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 + caddc 11113 · cmul 11115 − cmin 11444 -cneg 11445 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ∥ cdvds 16197 X_rm crmx 41638 Y_rm crmy 41639

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-dvds 16198 df-gcd 16436 df-numer 16671 df-denom 16672 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-squarenn 41579 df-pell1qr 41580 df-pell14qr 41581 df-pell1234qr 41582 df-pellfund 41583 df-rmx 41640 df-rmy 41641

This theorem is referenced by: jm2.26 41741

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