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Theorem ply1divmo 26041

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 22132	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	2, 4	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Ring)
6		ringgrp 20147	. . . . . . . . . . 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 20159	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
16	5, 11, 12, 15	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ 𝑞) ∈ 𝐵)
17		ply1divalg.m	. . . . . . . . . . . 12 ⊢ − = (-_g‘𝑃)
18	13, 17	grpsubcl 18952	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ (𝐺 ∙ 𝑞) ∈ 𝐵) → (𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵)
19	7, 9, 16, 18	syl3anc 1373	. . . . . . . . . 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 20159	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵) → (𝐺 ∙ 𝑟) ∈ 𝐵)
22	5, 11, 20, 21	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ 𝑟) ∈ 𝐵)
23	13, 17	grpsubcl 18952	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ (𝐺 ∙ 𝑟) ∈ 𝐵) → (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵)
24	7, 9, 22, 23	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵)
25	13, 17	grpsubcl 18952	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Grp ∧ (𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵 ∧ (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) ∈ 𝐵)
26	7, 19, 24, 25	syl3anc 1373	. . . . . . . . 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 25987	. . . . . . . . 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 25987	. . . . . . . . . 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 25987	. . . . . . . . . 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 4535	. . . . . . . 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 25987	. . . . . . . . 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 26012	. . . . . . . 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 13141	. . . . . . . . 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 1373	. . . . . . . 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 13116	. . . . . 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 25993	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ₀)
51	1, 10, 48, 50	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ₀)
52	51	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ∈ ℕ₀)
53	52	nn0red 12504	. . . . . . . . . 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 18952	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ Grp ∧ 𝑟 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑟 − 𝑞) ∈ 𝐵)
56	7, 20, 12, 55	syl3anc 1373	. . . . . . . . . . . 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 18958	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Grp ∧ 𝑟 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → ((𝑟 − 𝑞) = 0 ↔ 𝑟 = 𝑞))
59	7, 20, 12, 58	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) = 0 ↔ 𝑟 = 𝑞))
60		equcom 2018	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑞 ↔ 𝑞 = 𝑟)
61	59, 60	bitrdi 287	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) = 0 ↔ 𝑞 = 𝑟))
62	61	necon3bid 2969	. . . . . . . . . . . 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 25993	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑟 − 𝑞) ∈ 𝐵 ∧ (𝑟 − 𝑞) ≠ 0 ) → (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀)
65	54, 57, 63, 64	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀)
66		nn0addge1 12488	. . . . . . . . . 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 26019	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))) = ((𝐷‘𝐺) + (𝐷‘(𝑟 − 𝑞))))
74	67, 73	breqtrrd 5135	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
75		ringabl 20190	. . . . . . . . . . . . 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 19753	. . . . . . . . . . 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 20216	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ (𝑟 − 𝑞)) = ((𝐺 ∙ 𝑟) − (𝐺 ∙ 𝑞)))
79	77, 78	eqtr4d 2767	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) = (𝐺 ∙ (𝑟 − 𝑞)))
80	79	fveq2d 6862	. . . . . . . . 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 5135	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))))
83	36, 29	xrlenltd 11240	. . . . . . . 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 2944	. . . 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 3180	. 2 ⊢ (𝜑 → ∀𝑞 ∈ 𝐵 ∀𝑟 ∈ 𝐵 (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) → 𝑞 = 𝑟))
90		oveq2 7395	. . . . . 6 ⊢ (𝑞 = 𝑟 → (𝐺 ∙ 𝑞) = (𝐺 ∙ 𝑟))
91	90	oveq2d 7403	. . . . 5 ⊢ (𝑞 = 𝑟 → (𝐹 − (𝐺 ∙ 𝑞)) = (𝐹 − (𝐺 ∙ 𝑟)))
92	91	fveq2d 6862	. . . 4 ⊢ (𝑞 = 𝑟 → (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))))
93	92	breq1d 5117	. . 3 ⊢ (𝑞 = 𝑟 → ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
94	93	rmo4 3701	. 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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃*wrmo 3353 ifcif 4488 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 ℝcr 11067 + caddc 11071 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 ℕ₀cn0 12442 Basecbs 17179 ._rcmulr 17221 0_gc0g 17402 Grpcgrp 18865 -_gcsg 18867 Abelcabl 19711 Ringcrg 20142 RLRegcrlreg 20600 Poly₁cpl1 22061 coe₁cco1 22062 deg₁cdg1 25959

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-subrng 20455 df-subrg 20479 df-rlreg 20603 df-lmod 20768 df-lss 20838 df-cnfld 21265 df-psr 21818 df-mpl 21820 df-opsr 21822 df-psr1 22064 df-ply1 22066 df-coe1 22067 df-mdeg 25960 df-deg1 25961

This theorem is referenced by: ply1divalg 26043

W3C validator