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Theorem ply1divmo 26454

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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑅 ∈ Ring)
3		ply1divalg.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1ring 22565	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	2, 4	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Ring)
6		ringgrp 20464	. . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐹 ∈ 𝐵)
10		ply1divalg.g1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
11	10	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐺 ∈ 𝐵)
12		simprl 783	. . . . . . . . . . . 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 20477	. . . . . . . . . . . 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 19230	. . . . . . . . . . 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 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 ∈ 𝐵)
21	13, 14	ringcl 20477	. . . . . . . . . . . 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 19230	. . . . . . . . . . 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 19230	. . . . . . . . . 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 26400	. . . . . . . . 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 26400	. . . . . . . . . 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 26400	. . . . . . . . . 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 4529	. . . . . . . 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 26400	. . . . . . . . 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 486	. . . . . 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 26425	. . . . . . . 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 486	. . . . . . 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 13311	. . . . . . . . 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 483	. . . . . . 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 521	. . . . . 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 13285	. . . . . 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 418	. . . 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 26406	. . . . . . . . . . . . 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 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ∈ ℕ₀)
53	52	nn0red 12668	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ∈ ℝ)
54	1	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝑅 ∈ Ring)
55	13, 17	grpsubcl 19230	. . . . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝑟 − 𝑞) ∈ 𝐵)
58	13, 49, 17	grpsubeq0 19236	. . . . . . . . . . . . . . 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 2051	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑞 ↔ 𝑞 = 𝑟)
61	59, 60	bitrdi 290	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) = 0 ↔ 𝑞 = 𝑟))
62	61	necon3bid 3000	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑟 − 𝑞) ≠ 0 ↔ 𝑞 ≠ 𝑟))
63	62	biimpar 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝑟 − 𝑞) ≠ 0 )
64	27, 3, 49, 13	deg1nn0cl 26406	. . . . . . . . . . 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 12652	. . . . . . . . . 10 ⊢ (((𝐷‘𝐺) ∈ ℝ ∧ (𝐷‘(𝑟 − 𝑞)) ∈ ℕ₀) → (𝐷‘𝐺) ≤ ((𝐷‘𝐺) + (𝐷‘(𝑟 − 𝑞))))
67	53, 65, 66	syl2anc 596	. . . . . . . . 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 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝐺 ∈ 𝐵)
70	48	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → 𝐺 ≠ 0 )
71		ply1divmo.g3	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐸)
72	71	ad2antrr 739	. . . . . . . . . 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 26432	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))) = ((𝐷‘𝐺) + (𝐷‘(𝑟 − 𝑞))))
74	67, 73	breqtrrd 5133	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
75		ringabl 20510	. . . . . . . . . . . . 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 20037	. . . . . . . . . . 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 20538	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐺 ∙ (𝑟 − 𝑞)) = ((𝐺 ∙ 𝑟) − (𝐺 ∙ 𝑞)))
79	77, 78	eqtr4d 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟))) = (𝐺 ∙ (𝑟 − 𝑞)))
80	79	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) = (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
81	80	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) = (𝐷‘(𝐺 ∙ (𝑟 − 𝑞))))
82	74, 81	breqtrrd 5133	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝑞 ≠ 𝑟) → (𝐷‘𝐺) ≤ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))))
83	36, 29	xrlenltd 11375	. . . . . . . 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 486	. . . . . . 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 418	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞 ≠ 𝑟 → ¬ (𝐷‘((𝐹 − (𝐺 ∙ 𝑞)) − (𝐹 − (𝐺 ∙ 𝑟)))) < (𝐷‘𝐺)))
87	86	necon4ad 2975	. . . 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 3206	. 2 ⊢ (𝜑 → ∀𝑞 ∈ 𝐵 ∀𝑟 ∈ 𝐵 (((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)) → 𝑞 = 𝑟))
90		oveq2 7428	. . . . . 6 ⊢ (𝑞 = 𝑟 → (𝐺 ∙ 𝑞) = (𝐺 ∙ 𝑟))
91	90	oveq2d 7436	. . . . 5 ⊢ (𝑞 = 𝑟 → (𝐹 − (𝐺 ∙ 𝑞)) = (𝐹 − (𝐺 ∙ 𝑟)))
92	91	fveq2d 6889	. . . 4 ⊢ (𝑞 = 𝑟 → (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) = (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))))
93	92	breq1d 5113	. . 3 ⊢ (𝑞 = 𝑟 → ((𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺 ∙ 𝑟))) < (𝐷‘𝐺)))
94	93	rmo4 3688	. 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 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃*wrmo 3365 ifcif 4482 class class class wbr 5103 ‘cfv 6538 (class class class)co 7420 ℝcr 11199 + caddc 11203 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 ℕ₀cn0 12606 Basecbs 17387 ._rcmulr 17429 0_gc0g 17610 Grpcgrp 19144 -_gcsg 19146 Abelcabl 19995 Ringcrg 20459 RLRegcrlreg 20943 Poly₁cpl1 22495 coe₁cco1 22496 deg₁cdg1 26372

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-fz 13640 df-fzo 13789 df-seq 14145 df-hash 14475 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-0g 17612 df-gsum 17613 df-prds 17618 df-pws 17620 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-ghm 19428 df-cntz 19531 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-subrng 20798 df-subrg 20822 df-rlreg 20946 df-lmod 21137 df-lss 21207 df-cnfld 21679 df-psr 22217 df-mpl 22219 df-opsr 22221 df-psr1 22498 df-ply1 22500 df-coe1 22501 df-mdeg 26373 df-deg1 26374

This theorem is used by: ply1divalg 26456

W3C validator