Theorem dvdsq1p

Description: Divisibility in a polynomial ring is witnessed by the quotient. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	dvdsq1p.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	dvdsq1p.d	⊢ ∥ = ( ∥r ‘ 𝑃 )
	dvdsq1p.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	dvdsq1p.c	⊢ 𝐶 = ( Unic1p ‘ 𝑅 )
	dvdsq1p.t	⊢ · = ( .r ‘ 𝑃 )
	dvdsq1p.q	⊢ 𝑄 = ( quot1p ‘ 𝑅 )
Assertion	dvdsq1p	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐺 ∥ 𝐹 ↔ 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvdsq1p.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		dvdsq1p.d	⊢ ∥ = ( ∥r ‘ 𝑃 )
3		dvdsq1p.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
4		dvdsq1p.c	⊢ 𝐶 = ( Unic1p ‘ 𝑅 )
5		dvdsq1p.t	⊢ · = ( .r ‘ 𝑃 )
6		dvdsq1p.q	⊢ 𝑄 = ( quot1p ‘ 𝑅 )
7	1 3 4	uc1pcl	⊢ ( 𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵 )
8	7	3ad2ant3	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐺 ∈ 𝐵 )
9	3 2 5	dvdsr2	⊢ ( 𝐺 ∈ 𝐵 → ( 𝐺 ∥ 𝐹 ↔ ∃ 𝑞 ∈ 𝐵 ( 𝑞 · 𝐺 ) = 𝐹 ) )
10	8 9	syl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐺 ∥ 𝐹 ↔ ∃ 𝑞 ∈ 𝐵 ( 𝑞 · 𝐺 ) = 𝐹 ) )
11		eqcom	⊢ ( ( 𝑞 · 𝐺 ) = 𝐹 ↔ 𝐹 = ( 𝑞 · 𝐺 ) )
12		simprr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → 𝐹 = ( 𝑞 · 𝐺 ) )
13		simprl	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → 𝑞 ∈ 𝐵 )
14		simpl1	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → 𝑅 ∈ Ring )
15	1	ply1ring	⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring )
16	14 15	syl	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → 𝑃 ∈ Ring )
17		ringgrp	⊢ ( 𝑃 ∈ Ring → 𝑃 ∈ Grp )
18	16 17	syl	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → 𝑃 ∈ Grp )
19		simpl2	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → 𝐹 ∈ 𝐵 )
20		simpr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → 𝑞 ∈ 𝐵 )
21	8	adantr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → 𝐺 ∈ 𝐵 )
22	3 5	ringcl	⊢ ( ( 𝑃 ∈ Ring ∧ 𝑞 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝑞 · 𝐺 ) ∈ 𝐵 )
23	16 20 21 22	syl3anc	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → ( 𝑞 · 𝐺 ) ∈ 𝐵 )
24		eqid	⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 )
25		eqid	⊢ ( -g ‘ 𝑃 ) = ( -g ‘ 𝑃 )
26	3 24 25	grpsubeq0	⊢ ( ( 𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ( 𝑞 · 𝐺 ) ∈ 𝐵 ) → ( ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) = ( 0g ‘ 𝑃 ) ↔ 𝐹 = ( 𝑞 · 𝐺 ) ) )
27	18 19 23 26	syl3anc	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → ( ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) = ( 0g ‘ 𝑃 ) ↔ 𝐹 = ( 𝑞 · 𝐺 ) ) )
28	27	biimprd	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → ( 𝐹 = ( 𝑞 · 𝐺 ) → ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) = ( 0g ‘ 𝑃 ) ) )
29	28	impr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) = ( 0g ‘ 𝑃 ) )
30	29	fveq2d	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) ) = ( ( deg1 ‘ 𝑅 ) ‘ ( 0g ‘ 𝑃 ) ) )
31		simpl1	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → 𝑅 ∈ Ring )
32		eqid	⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑅 )
33	32 1 24	deg1z	⊢ ( 𝑅 ∈ Ring → ( ( deg1 ‘ 𝑅 ) ‘ ( 0g ‘ 𝑃 ) ) = -∞ )
34	31 33	syl	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 0g ‘ 𝑃 ) ) = -∞ )
35	30 34	eqtrd	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) ) = -∞ )
36	32 4	uc1pdeg	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐶 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) ∈ ℕ0 )
37	36	3adant2	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) ∈ ℕ0 )
38	37	nn0red	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) ∈ ℝ )
39	38	adantr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) ∈ ℝ )
40	39	mnfltd	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → -∞ < ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) )
41	35 40	eqbrtrd	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) ) < ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) )
42	6 1 3 32 25 5 4	q1peqb	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( 𝑞 ∈ 𝐵 ∧ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) ) < ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) ) ↔ ( 𝐹 𝑄 𝐺 ) = 𝑞 ) )
43	42	adantr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( ( 𝑞 ∈ 𝐵 ∧ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( 𝑞 · 𝐺 ) ) ) < ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) ) ↔ ( 𝐹 𝑄 𝐺 ) = 𝑞 ) )
44	13 41 43	mpbi2and	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( 𝐹 𝑄 𝐺 ) = 𝑞 )
45	44	oveq1d	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) = ( 𝑞 · 𝐺 ) )
46	12 45	eqtr4d	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ ( 𝑞 ∈ 𝐵 ∧ 𝐹 = ( 𝑞 · 𝐺 ) ) ) → 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) )
47	46	expr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → ( 𝐹 = ( 𝑞 · 𝐺 ) → 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )
48	11 47	biimtrid	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) ∧ 𝑞 ∈ 𝐵 ) → ( ( 𝑞 · 𝐺 ) = 𝐹 → 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )
49	48	rexlimdva	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ∃ 𝑞 ∈ 𝐵 ( 𝑞 · 𝐺 ) = 𝐹 → 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )
50	10 49	sylbid	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐺 ∥ 𝐹 → 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )
51	6 1 3 4	q1pcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝑄 𝐺 ) ∈ 𝐵 )
52	3 2 5	dvdsrmul	⊢ ( ( 𝐺 ∈ 𝐵 ∧ ( 𝐹 𝑄 𝐺 ) ∈ 𝐵 ) → 𝐺 ∥ ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) )
53	8 51 52	syl2anc	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐺 ∥ ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) )
54		breq2	⊢ ( 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) → ( 𝐺 ∥ 𝐹 ↔ 𝐺 ∥ ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )
55	53 54	syl5ibrcom	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) → 𝐺 ∥ 𝐹 ) )
56	50 55	impbid	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐺 ∥ 𝐹 ↔ 𝐹 = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )