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Theorem ply1divmo 26302

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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑅 ∈ Ring)
3		ply1divalg.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1ring 22416	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	2, 4	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Ring)
6		ringgrp 20324	. . . . . . . . . . 11 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
7	5, 6	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Grp)
8		ply1divalg.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
9	8	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐹 ∈ 𝐵)
10		ply1divalg.g1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
11	10	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐺 ∈ 𝐵)
12		simprl 782	. . . . . . . . . . . 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 20336	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺 ∙ 𝑞) ∈ 𝐵)
16	5, 11, 12, 15	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ 𝑞) ∈ 𝐵)
17		ply1divalg.m	. . . . . . . . . . . 12 ⊢ − = (-_g‘𝑃)
18	13, 17	grpsubcl 19090	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ (𝐺 ∙ 𝑞) ∈ 𝐵) → (𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵)
19	7, 9, 16, 18	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵)
20		simprr 784	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 ∈ 𝐵)
21	13, 14	ringcl 20336	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵) → (𝐺 ∙ 𝑟) ∈ 𝐵)
22	5, 11, 20, 21	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ 𝑟) ∈ 𝐵)
23	13, 17	grpsubcl 19090	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ (𝐺 ∙ 𝑟) ∈ 𝐵) → (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵)
24	7, 9, 22, 23	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵)
25	13, 17	grpsubcl 19090	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Grp ∧ (𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵 ∧ (𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) ∈ 𝐵)
26	7, 19, 24, 25	syl3anc 1398	. . . . . . . . 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 26248	. . . . . . . . 9 ⊢ (((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) ∈ 𝐵 → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ∈ ℝ^*)
29	26, 28	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) ∈ ℝ^*)
30	27, 3, 13	deg1xrcl 26248	. . . . . . . . . 10 ⊢ ((𝐹 − (𝐺 ∙ 𝑟)) ∈ 𝐵 → (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) ∈ ℝ^*)
31	24, 30	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) ∈ ℝ^*)
32	27, 3, 13	deg1xrcl 26248	. . . . . . . . . 10 ⊢ ((𝐹 − (𝐺 ∙ 𝑞)) ∈ 𝐵 → (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ∈ ℝ^*)
33	19, 32	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) ∈ ℝ^*)
34	31, 33	ifcld 4534	. . . . . . . 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 26248	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ^*)
36	11, 35	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘𝐺) ∈ ℝ^*)
37	29, 34, 36	3jca 1146	. . . . . . 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 485	. . . . . 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 26273	. . . . . . . 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 485	. . . . . . 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 13211	. . . . . . . . 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 1398	. . . . . . . 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 482	. . . . . . 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 520	. . . . . 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 13185	. . . . . 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 66	. . . . 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 417	. . . 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 26254	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ₀)
51	1, 10, 48, 50	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ₀)
52	51	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ∈ ℕ₀)
53	52	nn0red 12570	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ∈ ℝ)
54	1	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝑅 ∈ Ring)
55	13, 17	grpsubcl 19090	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ Grp ∧ 𝑟 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑟 − 𝑞) ∈ 𝐵)
56	7, 20, 12, 55	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑟 − 𝑞) ∈ 𝐵)
57	56	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝑟 − 𝑞) ∈ 𝐵)
58	13, 49, 17	grpsubeq0 19096	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Grp ∧ 𝑟 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → ((𝑟 − 𝑞) = 0 ↔ 𝑟 = 𝑞))
59	7, 20, 12, 58	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) = 0 ↔ 𝑟 = 𝑞))
60		equcom 2048	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑞 ↔ 𝑞 = 𝑟)
61	59, 60	bitrdi 290	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) = 0 ↔ 𝑞 = 𝑟))
62	61	necon3bid 3002	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) ≠ 0 ↔ 𝑞 ≠ 𝑟))
63	62	biimpar 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝑟 − 𝑞) ≠ 0 )
64	27, 3, 49, 13	deg1nn0cl 26254	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑟 − 𝑞) ∈ 𝐵 ∧ (𝑟 − 𝑞) ≠ 0 ) → (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀)
65	54, 57, 63, 64	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀)
66		nn0addge1 12554	. . . . . . . . . 10 ⊢ (((𝐷‘𝐺) ∈ ℝ ∧ (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀) → (𝐷‘𝐺) ≤ ((𝐷‘𝐺) + (𝐷‘(𝑟 − 𝑞))))
67	53, 65, 66	syl2anc 595	. . . . . . . . 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 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝐺 ∈ 𝐵)
70	48	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝐺 ≠ 0 )
71		ply1divmo.g3	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐸)
72	71	ad2antrr 738	. . . . . . . . . 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 26280	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))) = ((𝐷‘𝐺) + (𝐷‘(𝑟 − 𝑞))))
74	67, 73	breqtrrd 5139	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
75		ringabl 20369	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel)
76	5, 75	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Abel)
77	13, 17, 76, 9, 16, 22	ablnnncan1 19897	. . . . . . . . . . 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 20395	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ (𝑟 − 𝑞)) = ((𝐺 ∙ 𝑟) − (𝐺 ∙ 𝑞)))
79	77, 78	eqtr4d 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) = (𝐺 ∙ (𝑟 − 𝑞)))
80	79	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) = (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
81	80	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) = (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
82	74, 81	breqtrrd 5139	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))))
83	36, 29	xrlenltd 11279	. . . . . . . 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 485	. . . . . . 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 235	. . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → ¬ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺))
86	85	ex 417	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞 ≠ 𝑟 → ¬ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺)))
87	86	necon4ad 2977	. . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺) → 𝑞 = 𝑟))
88	47, 87	syld 48	. . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) → 𝑞 = 𝑟))
89	88	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑞 ∈ 𝐵 ∀𝑟 ∈ 𝐵 (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) → 𝑞 = 𝑟))
90		oveq2 7418	. . . . . 6 ⊢ (𝑞 = 𝑟 → (𝐺 ∙ 𝑞) = (𝐺 ∙ 𝑟))
91	90	oveq2d 7426	. . . . 5 ⊢ (𝑞 = 𝑟 → (𝐹 − (𝐺 ∙ 𝑞)) = (𝐹 − (𝐺 ∙ 𝑟)))
92	91	fveq2d 6885	. . . 4 ⊢ (𝑞 = 𝑟 → (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))))
93	92	breq1d 5119	. . 3 ⊢ (𝑞 = 𝑟 → ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
94	93	rmo4 3693	. 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 237	1 ⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃*wrmo 3368 ifcif 4487 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ℝcr 11103 + caddc 11107 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 ℕ₀cn0 12508 Basecbs 17273 ._rcmulr 17315 0_gc0g 17496 Grpcgrp 19004 -_gcsg 19006 Abelcabl 19855 Ringcrg 20319 RLRegcrlreg 20799 Poly₁cpl1 22346 coe₁cco1 22347 deg₁cdg1 26220

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-fz 13540 df-fzo 13688 df-seq 14043 df-hash 14372 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-hom 17338 df-cco 17339 df-0g 17498 df-gsum 17499 df-prds 17504 df-pws 17506 df-mre 17642 df-mrc 17643 df-acs 17645 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-mhm 18845 df-submnd 18846 df-grp 19007 df-minusg 19008 df-sbg 19009 df-mulg 19138 df-subg 19193 df-ghm 19288 df-cntz 19391 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-cring 20322 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-subrng 20654 df-subrg 20678 df-rlreg 20802 df-lmod 20992 df-lss 21062 df-cnfld 21532 df-psr 22068 df-mpl 22070 df-opsr 22072 df-psr1 22349 df-ply1 22351 df-coe1 22352 df-mdeg 26221 df-deg1 26222

This theorem is used by: ply1divalg 26304

W3C validator