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Theorem ply1divmo 26176

Description: Uniqueness of a quotient in a polynomial division. For polynomials 𝐹, 𝐺 such that 𝐺 ≠ 0 and the leading coefficient of 𝐺 is not a zero divisor, there is at most one polynomial 𝑞 which satisfies 𝐹 = (𝐺 · 𝑞) + 𝑟 where the degree of 𝑟 is less than the degree of 𝐺. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by NM, 17-Jun-2017.)

**Hypotheses**
Ref	Expression
ply1divalg.p	⊢ 𝑃 = (Poly₁‘𝑅)
ply1divalg.d	⊢ 𝐷 = (deg₁‘𝑅)
ply1divalg.b	⊢ 𝐵 = (Base‘𝑃)
ply1divalg.m	⊢ − = (-_g‘𝑃)
ply1divalg.z	⊢ 0 = (0_g‘𝑃)
ply1divalg.t	⊢ ∙ = (._r‘𝑃)
ply1divalg.r1	⊢ (𝜑 → 𝑅 ∈ Ring)
ply1divalg.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
ply1divalg.g1	⊢ (𝜑 → 𝐺 ∈ 𝐵)
ply1divalg.g2	⊢ (𝜑 → 𝐺 ≠ 0 )
ply1divmo.g3	⊢ (𝜑 → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐸)
ply1divmo.e	⊢ 𝐸 = (RLReg‘𝑅)

**Assertion**
Ref	Expression
ply1divmo	⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))

Distinct variable groups: 𝜑,𝑞 𝐵,𝑞 𝐷,𝑞 𝐹,𝑞 𝐺,𝑞 − ,𝑞 ∙ ,𝑞 Allowed substitution hints: 𝑃(𝑞) 𝑅(𝑞) 𝐸(𝑞) 0 (𝑞)

Proof of Theorem ply1divmo Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1divalg.r1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
2	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑅 ∈ Ring)
3		ply1divalg.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1ring 22250	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	2, 4	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Ring)
6		ringgrp 20236	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Grp)
8		ply1divalg.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐹 ∈ 𝐵)
10		ply1divalg.g1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
11	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐺 ∈ 𝐵)
12		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞 ∈ 𝐵)
13		ply1divalg.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑃)
14		ply1divalg.t	. . . . . . . . . . . . 13 ⊢ ∙ = (._r‘𝑃)
15	13, 14	ringcl 20248	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
16	5, 11, 12, 15	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ 𝑞) ∈ 𝐵)
17		ply1divalg.m	. . . . . . . . . . . 12 ⊢ − = (-_g‘𝑃)
18	13, 17	grpsubcl 19039	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ (𝐺 ∙ 𝑞) ∈ 𝐵) → (𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵)
19	7, 9, 16, 18	syl3anc 1372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵)
20		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 ∈ 𝐵)
21	13, 14	ringcl 20248	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵) → (𝐺 ∙ 𝑟) ∈ 𝐵)
22	5, 11, 20, 21	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ 𝑟) ∈ 𝐵)
23	13, 17	grpsubcl 19039	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ (𝐺 ∙ 𝑟) ∈ 𝐵) → (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵)
24	7, 9, 22, 23	syl3anc 1372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵)
25	13, 17	grpsubcl 19039	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Grp ∧ (𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵 ∧ (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) ∈ 𝐵)
26	7, 19, 24, 25	syl3anc 1372	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) ∈ 𝐵)
27		ply1divalg.d	. . . . . . . . . 10 ⊢ 𝐷 = (deg₁‘𝑅)
28	27, 3, 13	deg1xrcl 26122	. . . . . . . . 9 ⊢ (((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) ∈ 𝐵 → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ∈ ℝ^*)
29	26, 28	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ∈ ℝ^*)
30	27, 3, 13	deg1xrcl 26122	. . . . . . . . . 10 ⊢ ((𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵 → (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) ∈ ℝ^*)
31	24, 30	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) ∈ ℝ^*)
32	27, 3, 13	deg1xrcl 26122	. . . . . . . . . 10 ⊢ ((𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵 → (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ∈ ℝ^*)
33	19, 32	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ∈ ℝ^*)
34	31, 33	ifcld 4571	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) ∈ ℝ^*)
35	27, 3, 13	deg1xrcl 26122	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ^*)
36	11, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘𝐺) ∈ ℝ^*)
37	29, 34, 36	3jca 1128	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ∈ ℝ^* ∧ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) ∈ ℝ^* ∧ (𝐷‘𝐺) ∈ ℝ^*))
38	37	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))) → ((𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ∈ ℝ^* ∧ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) ∈ ℝ^* ∧ (𝐷‘𝐺) ∈ ℝ^*))
39	3, 27, 2, 13, 17, 19, 24	deg1suble 26147	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ≤ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))))
40	39	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ≤ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))))
41		xrmaxlt 13224	. . . . . . . . 9 ⊢ (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ∈ ℝ^* ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) ∈ ℝ^* ∧ (𝐷‘𝐺) ∈ ℝ^*) → (if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) < (𝐷‘𝐺) ↔ ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))))
42	33, 31, 36, 41	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) < (𝐷‘𝐺) ↔ ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))))
43	42	biimpar 477	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))) → if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) < (𝐷‘𝐺))
44	40, 43	jca 511	. . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))) → ((𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ≤ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) ∧ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) < (𝐷‘𝐺)))
45		xrlelttr 13199	. . . . . 6 ⊢ (((𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ∈ ℝ^* ∧ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) ∈ ℝ^* ∧ (𝐷‘𝐺) ∈ ℝ^*) → (((𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ≤ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) ∧ if((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ≤ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))), (𝐷‘(𝐹 − (𝐺 ∙ 𝑞)))) < (𝐷‘𝐺)) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺)))
46	38, 44, 45	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺))) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺))
47	46	ex 412	. . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺)))
48		ply1divalg.g2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ≠ 0 )
49		ply1divalg.z	. . . . . . . . . . . . . 14 ⊢ 0 = (0_g‘𝑃)
50	27, 3, 49, 13	deg1nn0cl 26128	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ₀)
51	1, 10, 48, 50	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ₀)
52	51	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ∈ ℕ₀)
53	52	nn0red 12590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ∈ ℝ)
54	1	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝑅 ∈ Ring)
55	13, 17	grpsubcl 19039	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ Grp ∧ 𝑟 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑟 − 𝑞) ∈ 𝐵)
56	7, 20, 12, 55	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑟 − 𝑞) ∈ 𝐵)
57	56	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝑟 − 𝑞) ∈ 𝐵)
58	13, 49, 17	grpsubeq0 19045	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Grp ∧ 𝑟 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → ((𝑟 − 𝑞) = 0 ↔ 𝑟 = 𝑞))
59	7, 20, 12, 58	syl3anc 1372	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) = 0 ↔ 𝑟 = 𝑞))
60		equcom 2016	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑞 ↔ 𝑞 = 𝑟)
61	59, 60	bitrdi 287	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) = 0 ↔ 𝑞 = 𝑟))
62	61	necon3bid 2984	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) ≠ 0 ↔ 𝑞 ≠ 𝑟))
63	62	biimpar 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝑟 − 𝑞) ≠ 0 )
64	27, 3, 49, 13	deg1nn0cl 26128	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑟 − 𝑞) ∈ 𝐵 ∧ (𝑟 − 𝑞) ≠ 0 ) → (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀)
65	54, 57, 63, 64	syl3anc 1372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀)
66		nn0addge1 12574	. . . . . . . . . 10 ⊢ (((𝐷‘𝐺) ∈ ℝ ∧ (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀) → (𝐷‘𝐺) ≤ ((𝐷‘𝐺) + (𝐷‘(𝑟 − 𝑞))))
67	53, 65, 66	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ ((𝐷‘𝐺) + (𝐷‘(𝑟 − 𝑞))))
68		ply1divmo.e	. . . . . . . . . 10 ⊢ 𝐸 = (RLReg‘𝑅)
69	10	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝐺 ∈ 𝐵)
70	48	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝐺 ≠ 0 )
71		ply1divmo.g3	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐸)
72	71	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐸)
73	27, 3, 68, 13, 14, 49, 54, 69, 70, 72, 57, 63	deg1mul2 26154	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))) = ((𝐷‘𝐺) + (𝐷‘(𝑟 − 𝑞))))
74	67, 73	breqtrrd 5170	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
75		ringabl 20279	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel)
76	5, 75	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Abel)
77	13, 17, 76, 9, 16, 22	ablnnncan1 19842	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) = ((𝐺 ∙ 𝑟) − (𝐺 ∙ 𝑞)))
78	13, 14, 17, 5, 11, 20, 12	ringsubdi 20305	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ (𝑟 − 𝑞)) = ((𝐺 ∙ 𝑟) − (𝐺 ∙ 𝑞)))
79	77, 78	eqtr4d 2779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) = (𝐺 ∙ (𝑟 − 𝑞)))
80	79	fveq2d 6909	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) = (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
81	80	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) = (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
82	74, 81	breqtrrd 5170	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))))
83	36, 29	xrlenltd 11328	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐷‘𝐺) ≤ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ↔ ¬ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺)))
84	83	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → ((𝐷‘𝐺) ≤ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ↔ ¬ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺)))
85	82, 84	mpbid 232	. . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → ¬ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺))
86	85	ex 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞 ≠ 𝑟 → ¬ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺)))
87	86	necon4ad 2958	. . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺) → 𝑞 = 𝑟))
88	47, 87	syld 47	. . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) → 𝑞 = 𝑟))
89	88	ralrimivva 3201	. 2 ⊢ (𝜑 → ∀𝑞 ∈ 𝐵 ∀𝑟 ∈ 𝐵 (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) → 𝑞 = 𝑟))
90		oveq2 7440	. . . . . 6 ⊢ (𝑞 = 𝑟 → (𝐺 ∙ 𝑞) = (𝐺 ∙ 𝑟))
91	90	oveq2d 7448	. . . . 5 ⊢ (𝑞 = 𝑟 → (𝐹 − (𝐺 ∙ 𝑞)) = (𝐹 − (𝐺 ∙ 𝑟)))
92	91	fveq2d 6909	. . . 4 ⊢ (𝑞 = 𝑟 → (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))))
93	92	breq1d 5152	. . 3 ⊢ (𝑞 = 𝑟 → ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
94	93	rmo4 3735	. 2 ⊢ (∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ ∀𝑞 ∈ 𝐵 ∀𝑟 ∈ 𝐵 (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) → 𝑞 = 𝑟))
95	89, 94	sylibr 234	1 ⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃*wrmo 3378 ifcif 4524 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 + caddc 11159 ℝ^*cxr 11295 < clt 11296 ≤ cle 11297 ℕ₀cn0 12528 Basecbs 17248 ._rcmulr 17299 0_gc0g 17485 Grpcgrp 18952 -_gcsg 18954 Abelcabl 19800 Ringcrg 20231 RLRegcrlreg 20692 Poly₁cpl1 22179 coe₁cco1 22180 deg₁cdg1 26094

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-subrng 20547 df-subrg 20571 df-rlreg 20695 df-lmod 20861 df-lss 20931 df-cnfld 21366 df-psr 21930 df-mpl 21932 df-opsr 21934 df-psr1 22182 df-ply1 22184 df-coe1 22185 df-mdeg 26095 df-deg1 26096

This theorem is referenced by: ply1divalg 26178

W3C validator