aks6d1c5lem1 - Mathbox for metakunt

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Theorem aks6d1c5lem1 43323

Description: Lemma for claim 5, evaluate the linear factor at -c to get a root. (Contributed by metakunt, 5-May-2025.)

**Hypotheses**
Ref	Expression
aks6d1p5.1	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1p5.2	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c5.3	⊢ 𝑃 = (chr‘𝐾)
aks6d1c5.4	⊢ (𝜑 → 𝐴 ∈ ℕ₀)
aks6d1c5.5	⊢ (𝜑 → 𝐴 < 𝑃)
aks6d1c5.6	⊢ 𝑋 = (var₁‘𝐾)
aks6d1c5.7	⊢ ↑ = (._g‘(mulGrp‘(Poly₁‘𝐾)))
aks6d1c5.8	⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c5p1.1	⊢ (𝜑 → 𝐵 ∈ (0...𝐴))
aks6d1c5p1.2	⊢ (𝜑 → 𝐶 ∈ (0...𝐴))

**Assertion**
Ref	Expression
aks6d1c5lem1	⊢ (𝜑 → (𝐵 = 𝐶 ↔ (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵))))‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = (0_g‘𝐾)))

**Proof of Theorem aks6d1c5lem1**
Step	Hyp	Ref	Expression
1		zringplusg 21769	. . . . . . . . . . 11 ⊢ + = (+_g‘ℤ_ring)
2	1	eqcomi 2769	. . . . . . . . . 10 ⊢ (+_g‘ℤ_ring) = +
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘ℤ_ring) = + )
4	3	oveqd 7441	. . . . . . . 8 ⊢ (𝜑 → ((0 − 𝐶)(+_g‘ℤ_ring)𝐵) = ((0 − 𝐶) + 𝐵))
5		0cnd 11306	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ)
6		aks6d1c5p1.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ (0...𝐴))
7	6	elfzelzd 13664	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℤ)
8	7	zcnd 12811	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ)
9		aks6d1c5p1.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (0...𝐴))
10	9	elfzelzd 13664	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ)
11	10	zcnd 12811	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
12	5, 8, 11	subadd23d 11698	. . . . . . . . 9 ⊢ (𝜑 → ((0 − 𝐶) + 𝐵) = (0 + (𝐵 − 𝐶)))
13	11, 8	subcld 11676	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐶) ∈ ℂ)
14	13	addlidd 11518	. . . . . . . . 9 ⊢ (𝜑 → (0 + (𝐵 − 𝐶)) = (𝐵 − 𝐶))
15	12, 14	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → ((0 − 𝐶) + 𝐵) = (𝐵 − 𝐶))
16	4, 15	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → ((0 − 𝐶)(+_g‘ℤ_ring)𝐵) = (𝐵 − 𝐶))
17	16	fveq2d 6894	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = ((ℤRHom‘𝐾)‘(𝐵 − 𝐶)))
18	17	eqeq1d 2762	. . . . 5 ⊢ (𝜑 → (((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (0_g‘𝐾) ↔ ((ℤRHom‘𝐾)‘(𝐵 − 𝐶)) = (0_g‘𝐾)))
19		aks6d1p5.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℙ)
20	19	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∈ ℙ)
21		prmnn 16858	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
22	20, 21	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∈ ℕ)
23	22	nnzd 12726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∈ ℤ)
24		dvds0 16450	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℤ → 𝑃 ∥ 0)
25	23, 24	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∥ 0)
26	11	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ ℂ)
27	26	subidd 11664	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵 − 𝐵) = 0)
28	27	eqcomd 2766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 0 = (𝐵 − 𝐵))
29		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
30	29	oveq2d 7440	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵 − 𝐵) = (𝐵 − 𝐶))
31	28, 30	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 0 = (𝐵 − 𝐶))
32	25, 31	breqtrd 5130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∥ (𝐵 − 𝐶))
33	32	ex 418	. . . . . . 7 ⊢ (𝜑 → (𝐵 = 𝐶 → 𝑃 ∥ (𝐵 − 𝐶)))
34	19, 21	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ)
35	34	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 𝑃 ∈ ℕ)
36	35	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝑃 ∈ ℕ)
37		1zzd 12734	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 1 ∈ ℤ)
38	36	nnzd 12726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝑃 ∈ ℤ)
39	38, 37	zsubcld 12815	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝑃 − 1) ∈ ℤ)
40	10, 7	zsubcld 12815	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐶) ∈ ℤ)
41	40	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ∈ ℤ)
42		1e0p1 12868	. . . . . . . . . . . . . 14 ⊢ 1 = (0 + 1)
43	42	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 1 = (0 + 1))
44		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝐶 < 𝐵)
45	7	zred 12810	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ℝ)
46	45	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 𝐶 ∈ ℝ)
47	46	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝐶 ∈ ℝ)
48	10	zred 12810	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ)
49	48	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 𝐵 ∈ ℝ)
50	49	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝐵 ∈ ℝ)
51	47, 50	posdifd 11910	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐶 < 𝐵 ↔ 0 < (𝐵 − 𝐶)))
52	44, 51	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 0 < (𝐵 − 𝐶))
53		0zd 12712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 0 ∈ ℤ)
54	53, 41	zltp1led 12758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (0 < (𝐵 − 𝐶) ↔ (0 + 1) ≤ (𝐵 − 𝐶)))
55	52, 54	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (0 + 1) ≤ (𝐵 − 𝐶))
56	43, 55	eqbrtrd 5126	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 1 ≤ (𝐵 − 𝐶))
57	41	zred 12810	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ∈ ℝ)
58	36	nnred 12357	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝑃 ∈ ℝ)
59		elfzle1 13667	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
60	6, 59	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ 𝐶)
61	60	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 0 ≤ 𝐶)
62	61	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 0 ≤ 𝐶)
63	50, 47	subge02d 11915	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (0 ≤ 𝐶 ↔ (𝐵 − 𝐶) ≤ 𝐵))
64	62, 63	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ≤ 𝐵)
65		aks6d1c5.4	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
66	65	nn0red 12675	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ)
67	34	nnred 12357	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℝ)
68		elfzle2 13668	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ (0...𝐴) → 𝐵 ≤ 𝐴)
69	9, 68	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ≤ 𝐴)
70		aks6d1c5.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 < 𝑃)
71	48, 66, 67, 69, 70	lelttrd 11475	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 < 𝑃)
72	71	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 𝐵 < 𝑃)
73	72	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝐵 < 𝑃)
74	57, 50, 58, 64, 73	lelttrd 11475	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) < 𝑃)
75	41, 38	zltlem1d 12757	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → ((𝐵 − 𝐶) < 𝑃 ↔ (𝐵 − 𝐶) ≤ (𝑃 − 1)))
76	74, 75	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ≤ (𝑃 − 1))
77	37, 39, 41, 56, 76	elfzd 13654	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ∈ (1...(𝑃 − 1)))
78		fzm1ndvds 16501	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℕ ∧ (𝐵 − 𝐶) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
79	36, 77, 78	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
80		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ ¬ 𝐶 < 𝐵) → 𝜑)
81		axlttri 11388	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶 < 𝐵)))
82	48, 45, 81	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶 < 𝐵)))
83		ioran 999	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐵 = 𝐶 ∨ 𝐶 < 𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 < 𝐵))
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (¬ (𝐵 = 𝐶 ∨ 𝐶 < 𝐵) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 < 𝐵)))
85	82, 84	bitr2d 283	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 < 𝐵) ↔ 𝐵 < 𝐶))
86	85	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 < 𝐵) → 𝐵 < 𝐶))
87	86	imp 412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 < 𝐵)) → 𝐵 < 𝐶)
88	87	anassrs 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ ¬ 𝐶 < 𝐵) → 𝐵 < 𝐶)
89	80, 88	jca 521	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ ¬ 𝐶 < 𝐵) → (𝜑 ∧ 𝐵 < 𝐶))
90	34	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝑃 ∈ ℕ)
91		1zzd 12734	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 1 ∈ ℤ)
92	34	nnzd 12726	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℤ)
93	92	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝑃 ∈ ℤ)
94	93, 91	zsubcld 12815	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝑃 − 1) ∈ ℤ)
95	7	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℤ)
96	10	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℤ)
97	95, 96	zsubcld 12815	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ∈ ℤ)
98	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 1 = (0 + 1))
99	48, 45	posdifd 11910	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ 0 < (𝐶 − 𝐵)))
100	99	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 < 𝐶 → 0 < (𝐶 − 𝐵)))
101	100	imp 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 0 < (𝐶 − 𝐵))
102		0zd 12712	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 0 ∈ ℤ)
103	102, 97	zltp1led 12758	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (0 < (𝐶 − 𝐵) ↔ (0 + 1) ≤ (𝐶 − 𝐵)))
104	101, 103	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (0 + 1) ≤ (𝐶 − 𝐵))
105	98, 104	eqbrtrd 5126	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 1 ≤ (𝐶 − 𝐵))
106	97	zred 12810	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ∈ ℝ)
107	45	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℝ)
108	67	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝑃 ∈ ℝ)
109	9	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐵 ∈ (0...𝐴))
110		elfzle1 13667	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ (0...𝐴) → 0 ≤ 𝐵)
111	109, 110	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 0 ≤ 𝐵)
112	48	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ)
113	107, 112	subge02d 11915	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (0 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ 𝐶))
114	111, 113	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ≤ 𝐶)
115	66	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℝ)
116		elfzle2 13668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ≤ 𝐴)
117	6, 116	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ≤ 𝐴)
118	117	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐶 ≤ 𝐴)
119	70	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐴 < 𝑃)
120	107, 115, 108, 118, 119	lelttrd 11475	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐶 < 𝑃)
121	106, 107, 108, 114, 120	lelttrd 11475	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) < 𝑃)
122	97, 93	zltlem1d 12757	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → ((𝐶 − 𝐵) < 𝑃 ↔ (𝐶 − 𝐵) ≤ (𝑃 − 1)))
123	121, 122	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ≤ (𝑃 − 1))
124	91, 94, 97, 105, 123	elfzd 13654	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ∈ (1...(𝑃 − 1)))
125		fzm1ndvds 16501	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℕ ∧ (𝐶 − 𝐵) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (𝐶 − 𝐵))
126	90, 124, 125	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → ¬ 𝑃 ∥ (𝐶 − 𝐵))
127		dvdsnegb 16452	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ (𝐵 − 𝐶) ∈ ℤ) → (𝑃 ∥ (𝐵 − 𝐶) ↔ 𝑃 ∥ -(𝐵 − 𝐶)))
128	92, 40, 127	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∥ (𝐵 − 𝐶) ↔ 𝑃 ∥ -(𝐵 − 𝐶)))
129	11, 8	negsubdi2d 11692	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → -(𝐵 − 𝐶) = (𝐶 − 𝐵))
130	129	breq2d 5114	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∥ -(𝐵 − 𝐶) ↔ 𝑃 ∥ (𝐶 − 𝐵)))
131	128, 130	bitrd 282	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 ∥ (𝐵 − 𝐶) ↔ 𝑃 ∥ (𝐶 − 𝐵)))
132	131	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝑃 ∥ (𝐵 − 𝐶) ↔ 𝑃 ∥ (𝐶 − 𝐵)))
133	126, 132	mtbird 328	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
134	89, 133	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ ¬ 𝐶 < 𝐵) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
135	79, 134	pm2.61dan 825	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
136	135	ex 418	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝐵 = 𝐶 → ¬ 𝑃 ∥ (𝐵 − 𝐶)))
137	136	con4d 116	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∥ (𝐵 − 𝐶) → 𝐵 = 𝐶))
138	33, 137	impbid 215	. . . . . 6 ⊢ (𝜑 → (𝐵 = 𝐶 ↔ 𝑃 ∥ (𝐵 − 𝐶)))
139		aks6d1p5.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Field)
140	139	fldcrngd 21004	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
141		crngring 20481	. . . . . . . 8 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
142	140, 141	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Ring)
143		aks6d1c5.3	. . . . . . . 8 ⊢ 𝑃 = (chr‘𝐾)
144		eqid 2760	. . . . . . . 8 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
145		eqid 2760	. . . . . . . 8 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
146	143, 144, 145	chrdvds 21841	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ (𝐵 − 𝐶) ∈ ℤ) → (𝑃 ∥ (𝐵 − 𝐶) ↔ ((ℤRHom‘𝐾)‘(𝐵 − 𝐶)) = (0_g‘𝐾)))
147	142, 40, 146	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ (𝐵 − 𝐶) ↔ ((ℤRHom‘𝐾)‘(𝐵 − 𝐶)) = (0_g‘𝐾)))
148	138, 147	bitr2d 283	. . . . 5 ⊢ (𝜑 → (((ℤRHom‘𝐾)‘(𝐵 − 𝐶)) = (0_g‘𝐾) ↔ 𝐵 = 𝐶))
149	18, 148	bitrd 282	. . . 4 ⊢ (𝜑 → (((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (0_g‘𝐾) ↔ 𝐵 = 𝐶))
150	149	bicomd 226	. . 3 ⊢ (𝜑 → (𝐵 = 𝐶 ↔ ((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (0_g‘𝐾)))
151	144	zrhrhm 21826	. . . . . . 7 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
152		rhmghm 20723	. . . . . . 7 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
153	151, 152	syl 18	. . . . . 6 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
154	142, 153	syl 18	. . . . 5 ⊢ (𝜑 → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
155		0zd 12712	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
156	155, 7	zsubcld 12815	. . . . . 6 ⊢ (𝜑 → (0 − 𝐶) ∈ ℤ)
157		zringbas 21768	. . . . . 6 ⊢ ℤ = (Base‘ℤ_ring)
158	156, 157	eleqtrdi 2870	. . . . 5 ⊢ (𝜑 → (0 − 𝐶) ∈ (Base‘ℤ_ring))
159	10, 157	eleqtrdi 2870	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Base‘ℤ_ring))
160		eqid 2760	. . . . . 6 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
161		eqid 2760	. . . . . 6 ⊢ (+_g‘ℤ_ring) = (+_g‘ℤ_ring)
162		eqid 2760	. . . . . 6 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
163	160, 161, 162	ghmlin 19444	. . . . 5 ⊢ (((ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾) ∧ (0 − 𝐶) ∈ (Base‘ℤ_ring) ∧ 𝐵 ∈ (Base‘ℤ_ring)) → ((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)))
164	154, 158, 159, 163	syl3anc 1398	. . . 4 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)))
165	164	eqeq1d 2762	. . 3 ⊢ (𝜑 → (((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (0_g‘𝐾) ↔ (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)) = (0_g‘𝐾)))
166	150, 165	bitrd 282	. 2 ⊢ (𝜑 → (𝐵 = 𝐶 ↔ (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)) = (0_g‘𝐾)))
167		eqid 2760	. . . . . 6 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
168		eqid 2760	. . . . . 6 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
169		eqid 2760	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
170		eqid 2760	. . . . . 6 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
171	157, 169	ghmf 19443	. . . . . . . 8 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
172	154, 171	syl 18	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
173	172, 156	ffvelcdmd 7090	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘(0 − 𝐶)) ∈ (Base‘𝐾))
174		aks6d1c5.6	. . . . . . 7 ⊢ 𝑋 = (var₁‘𝐾)
175	167, 174, 169, 168, 170, 140, 173	evl1vard 22664	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = ((ℤRHom‘𝐾)‘(0 − 𝐶))))
176		eqid 2760	. . . . . . 7 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
177	172, 10	ffvelcdmd 7090	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐵) ∈ (Base‘𝐾))
178	167, 168, 169, 176, 170, 140, 177, 173	evl1scad 22662	. . . . . 6 ⊢ (𝜑 → (((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵)) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵)))‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = ((ℤRHom‘𝐾)‘𝐵)))
179		eqid 2760	. . . . . 6 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
180	167, 168, 169, 170, 140, 173, 175, 178, 179, 162	evl1addd 22668	. . . . 5 ⊢ (𝜑 → ((𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵))))‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵))))
181	180	simprd 501	. . . 4 ⊢ (𝜑 → (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵))))‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)))
182	181	eqcomd 2766	. . 3 ⊢ (𝜑 → (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)) = (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵))))‘((ℤRHom‘𝐾)‘(0 − 𝐶))))
183	182	eqeq1d 2762	. 2 ⊢ (𝜑 → ((((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)) = (0_g‘𝐾) ↔ (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵))))‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = (0_g‘𝐾)))
184	166, 183	bitrd 282	1 ⊢ (𝜑 → (𝐵 = 𝐶 ↔ (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝐵))))‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = (0_g‘𝐾)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 class class class wbr 5102 ↦ cmpt 5185 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ↑_m cmap 8855 ℂcc 11205 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 < clt 11350 ≤ cle 11351 − cmin 11548 -cneg 11549 ℕcn 12342 ℕ₀cn0 12613 ℤcz 12700 ...cfz 13646 ∥ cdvds 16431 ℙcprime 16855 Basecbs 17396 +_gcplusg 17437 0_gc0g 17619 Σ_g cgsu 17620 ._gcmg 19286 GrpHom cghm 19436 mulGrpcmgp 20369 Ringcrg 20468 CRingccrg 20469 RingHom crh 20708 Fieldcfield 20990 ℤ_ringczring 21761 ℤRHomczrh 21814 chrcchr 21816 algSccascl 22169 var₁cv1 22503 Poly₁cpl1 22504 eval₁ce1 22641

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286 ax-mulf 11287

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 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 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-ofr 7698 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-rp 13128 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-dvds 16432 df-prm 16856 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-0g 17621 df-gsum 17622 df-prds 17627 df-pws 17629 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-mhm 18987 df-submnd 18988 df-grp 19156 df-minusg 19157 df-sbg 19158 df-mulg 19287 df-subg 19342 df-ghm 19437 df-cntz 19540 df-od 19751 df-cmn 20005 df-abl 20006 df-mgp 20370 df-rng 20384 df-ur 20417 df-srg 20422 df-ring 20470 df-cring 20471 df-rhm 20711 df-subrng 20807 df-subrg 20831 df-field 20992 df-lmod 21146 df-lss 21216 df-lsp 21256 df-cnfld 21688 df-zring 21762 df-zrh 21818 df-chr 21820 df-assa 22170 df-asp 22171 df-ascl 22172 df-psr 22226 df-mvr 22227 df-mpl 22228 df-opsr 22230 df-evls 22392 df-evl 22393 df-psr1 22507 df-vr1 22508 df-ply1 22509 df-evl1 22643

This theorem is used by: aks6d1c5lem2 43325

W3C validator