aks6d1c5lem1 - Mathbox for metakunt

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Theorem aks6d1c5lem1 41639

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 21387	. . . . . . . . . . 11 ⊢ + = (+_g‘ℤ_ring)
2	1	eqcomi 2737	. . . . . . . . . 10 ⊢ (+_g‘ℤ_ring) = +
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘ℤ_ring) = + )
4	3	oveqd 7443	. . . . . . . 8 ⊢ (𝜑 → ((0 − 𝐶)(+_g‘ℤ_ring)𝐵) = ((0 − 𝐶) + 𝐵))
5		0cnd 11245	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℂ)
6		aks6d1c5p1.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ (0...𝐴))
7	6	elfzelzd 13542	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℤ)
8	7	zcnd 12705	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ)
9		aks6d1c5p1.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (0...𝐴))
10	9	elfzelzd 13542	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ)
11	10	zcnd 12705	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
12	5, 8, 11	subadd23d 11631	. . . . . . . . 9 ⊢ (𝜑 → ((0 − 𝐶) + 𝐵) = (0 + (𝐵 − 𝐶)))
13	11, 8	subcld 11609	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐶) ∈ ℂ)
14	13	addlidd 11453	. . . . . . . . 9 ⊢ (𝜑 → (0 + (𝐵 − 𝐶)) = (𝐵 − 𝐶))
15	12, 14	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → ((0 − 𝐶) + 𝐵) = (𝐵 − 𝐶))
16	4, 15	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → ((0 − 𝐶)(+_g‘ℤ_ring)𝐵) = (𝐵 − 𝐶))
17	16	fveq2d 6906	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = ((ℤRHom‘𝐾)‘(𝐵 − 𝐶)))
18	17	eqeq1d 2730	. . . . 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 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∈ ℙ)
21		prmnn 16652	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∈ ℕ)
23	22	nnzd 12623	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∈ ℤ)
24		dvds0 16256	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℤ → 𝑃 ∥ 0)
25	23, 24	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∥ 0)
26	11	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ ℂ)
27	26	subidd 11597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵 − 𝐵) = 0)
28	27	eqcomd 2734	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 0 = (𝐵 − 𝐵))
29		simpr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
30	29	oveq2d 7442	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵 − 𝐵) = (𝐵 − 𝐶))
31	28, 30	eqtrd 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 0 = (𝐵 − 𝐶))
32	25, 31	breqtrd 5178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝑃 ∥ (𝐵 − 𝐶))
33	32	ex 411	. . . . . . 7 ⊢ (𝜑 → (𝐵 = 𝐶 → 𝑃 ∥ (𝐵 − 𝐶)))
34	19, 21	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ)
35	34	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 𝑃 ∈ ℕ)
36	35	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝑃 ∈ ℕ)
37		1zzd 12631	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 1 ∈ ℤ)
38	36	nnzd 12623	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝑃 ∈ ℤ)
39	38, 37	zsubcld 12709	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝑃 − 1) ∈ ℤ)
40	10, 7	zsubcld 12709	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐶) ∈ ℤ)
41	40	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ∈ ℤ)
42		1e0p1 12757	. . . . . . . . . . . . . 14 ⊢ 1 = (0 + 1)
43	42	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 1 = (0 + 1))
44		simpr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝐶 < 𝐵)
45	7	zred 12704	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ℝ)
46	45	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 𝐶 ∈ ℝ)
47	46	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝐶 ∈ ℝ)
48	10	zred 12704	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ)
49	48	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 𝐵 ∈ ℝ)
50	49	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝐵 ∈ ℝ)
51	47, 50	posdifd 11839	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐶 < 𝐵 ↔ 0 < (𝐵 − 𝐶)))
52	44, 51	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 0 < (𝐵 − 𝐶))
53		0zd 12608	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 0 ∈ ℤ)
54	53, 41	zltp1led 41482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (0 < (𝐵 − 𝐶) ↔ (0 + 1) ≤ (𝐵 − 𝐶)))
55	52, 54	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (0 + 1) ≤ (𝐵 − 𝐶))
56	43, 55	eqbrtrd 5174	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 1 ≤ (𝐵 − 𝐶))
57	41	zred 12704	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ∈ ℝ)
58	36	nnred 12265	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝑃 ∈ ℝ)
59		elfzle1 13544	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ (0...𝐴) → 0 ≤ 𝐶)
60	6, 59	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ 𝐶)
61	60	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 0 ≤ 𝐶)
62	61	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 0 ≤ 𝐶)
63	50, 47	subge02d 11844	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (0 ≤ 𝐶 ↔ (𝐵 − 𝐶) ≤ 𝐵))
64	62, 63	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ≤ 𝐵)
65		aks6d1c5.4	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
66	65	nn0red 12571	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ)
67	34	nnred 12265	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℝ)
68		elfzle2 13545	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ (0...𝐴) → 𝐵 ≤ 𝐴)
69	9, 68	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ≤ 𝐴)
70		aks6d1c5.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 < 𝑃)
71	48, 66, 67, 69, 70	lelttrd 11410	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 < 𝑃)
72	71	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → 𝐵 < 𝑃)
73	72	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → 𝐵 < 𝑃)
74	57, 50, 58, 64, 73	lelttrd 11410	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) < 𝑃)
75	41, 38	zltlem1d 41481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → ((𝐵 − 𝐶) < 𝑃 ↔ (𝐵 − 𝐶) ≤ (𝑃 − 1)))
76	74, 75	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ≤ (𝑃 − 1))
77	37, 39, 41, 56, 76	elfzd 13532	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → (𝐵 − 𝐶) ∈ (1...(𝑃 − 1)))
78		fzm1ndvds 16306	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℕ ∧ (𝐵 − 𝐶) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
79	36, 77, 78	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ 𝐶 < 𝐵) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
80		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ ¬ 𝐶 < 𝐵) → 𝜑)
81		axlttri 11323	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶 < 𝐵)))
82	48, 45, 81	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶 < 𝐵)))
83		ioran 981	. . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 < 𝐵) ↔ 𝐵 < 𝐶))
86	85	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 < 𝐵) → 𝐵 < 𝐶))
87	86	imp 405	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 < 𝐵)) → 𝐵 < 𝐶)
88	87	anassrs 466	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ ¬ 𝐶 < 𝐵) → 𝐵 < 𝐶)
89	80, 88	jca 510	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ ¬ 𝐶 < 𝐵) → (𝜑 ∧ 𝐵 < 𝐶))
90	34	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝑃 ∈ ℕ)
91		1zzd 12631	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 1 ∈ ℤ)
92	34	nnzd 12623	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℤ)
93	92	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝑃 ∈ ℤ)
94	93, 91	zsubcld 12709	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝑃 − 1) ∈ ℤ)
95	7	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℤ)
96	10	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℤ)
97	95, 96	zsubcld 12709	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ∈ ℤ)
98	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 1 = (0 + 1))
99	48, 45	posdifd 11839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ 0 < (𝐶 − 𝐵)))
100	99	biimpd 228	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 < 𝐶 → 0 < (𝐶 − 𝐵)))
101	100	imp 405	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 0 < (𝐶 − 𝐵))
102		0zd 12608	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 0 ∈ ℤ)
103	102, 97	zltp1led 41482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (0 < (𝐶 − 𝐵) ↔ (0 + 1) ≤ (𝐶 − 𝐵)))
104	101, 103	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (0 + 1) ≤ (𝐶 − 𝐵))
105	98, 104	eqbrtrd 5174	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 1 ≤ (𝐶 − 𝐵))
106	97	zred 12704	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ∈ ℝ)
107	45	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℝ)
108	67	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝑃 ∈ ℝ)
109	9	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐵 ∈ (0...𝐴))
110		elfzle1 13544	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ (0...𝐴) → 0 ≤ 𝐵)
111	109, 110	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 0 ≤ 𝐵)
112	48	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ)
113	107, 112	subge02d 11844	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (0 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ 𝐶))
114	111, 113	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ≤ 𝐶)
115	66	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℝ)
116		elfzle2 13545	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ≤ 𝐴)
117	6, 116	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ≤ 𝐴)
118	117	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐶 ≤ 𝐴)
119	70	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐴 < 𝑃)
120	107, 115, 108, 118, 119	lelttrd 11410	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → 𝐶 < 𝑃)
121	106, 107, 108, 114, 120	lelttrd 11410	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) < 𝑃)
122	97, 93	zltlem1d 41481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → ((𝐶 − 𝐵) < 𝑃 ↔ (𝐶 − 𝐵) ≤ (𝑃 − 1)))
123	121, 122	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ≤ (𝑃 − 1))
124	91, 94, 97, 105, 123	elfzd 13532	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝐶 − 𝐵) ∈ (1...(𝑃 − 1)))
125		fzm1ndvds 16306	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℕ ∧ (𝐶 − 𝐵) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (𝐶 − 𝐵))
126	90, 124, 125	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → ¬ 𝑃 ∥ (𝐶 − 𝐵))
127		dvdsnegb 16258	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ (𝐵 − 𝐶) ∈ ℤ) → (𝑃 ∥ (𝐵 − 𝐶) ↔ 𝑃 ∥ -(𝐵 − 𝐶)))
128	92, 40, 127	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∥ (𝐵 − 𝐶) ↔ 𝑃 ∥ -(𝐵 − 𝐶)))
129	11, 8	negsubdi2d 11625	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → -(𝐵 − 𝐶) = (𝐶 − 𝐵))
130	129	breq2d 5164	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∥ -(𝐵 − 𝐶) ↔ 𝑃 ∥ (𝐶 − 𝐵)))
131	128, 130	bitrd 278	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 ∥ (𝐵 − 𝐶) ↔ 𝑃 ∥ (𝐶 − 𝐵)))
132	131	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → (𝑃 ∥ (𝐵 − 𝐶) ↔ 𝑃 ∥ (𝐶 − 𝐵)))
133	126, 132	mtbird 324	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 < 𝐶) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
134	89, 133	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐵 = 𝐶) ∧ ¬ 𝐶 < 𝐵) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
135	79, 134	pm2.61dan 811	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 = 𝐶) → ¬ 𝑃 ∥ (𝐵 − 𝐶))
136	135	ex 411	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝐵 = 𝐶 → ¬ 𝑃 ∥ (𝐵 − 𝐶)))
137	136	con4d 115	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∥ (𝐵 − 𝐶) → 𝐵 = 𝐶))
138	33, 137	impbid 211	. . . . . 6 ⊢ (𝜑 → (𝐵 = 𝐶 ↔ 𝑃 ∥ (𝐵 − 𝐶)))
139		aks6d1p5.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Field)
140	139	fldcrngd 20644	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ CRing)
141		crngring 20192	. . . . . . . 8 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
142	140, 141	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Ring)
143		aks6d1c5.3	. . . . . . . 8 ⊢ 𝑃 = (chr‘𝐾)
144		eqid 2728	. . . . . . . 8 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
145		eqid 2728	. . . . . . . 8 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
146	143, 144, 145	chrdvds 21463	. . . . . . 7 ⊢ ((𝐾 ∈ Ring ∧ (𝐵 − 𝐶) ∈ ℤ) → (𝑃 ∥ (𝐵 − 𝐶) ↔ ((ℤRHom‘𝐾)‘(𝐵 − 𝐶)) = (0_g‘𝐾)))
147	142, 40, 146	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ (𝐵 − 𝐶) ↔ ((ℤRHom‘𝐾)‘(𝐵 − 𝐶)) = (0_g‘𝐾)))
148	138, 147	bitr2d 279	. . . . 5 ⊢ (𝜑 → (((ℤRHom‘𝐾)‘(𝐵 − 𝐶)) = (0_g‘𝐾) ↔ 𝐵 = 𝐶))
149	18, 148	bitrd 278	. . . 4 ⊢ (𝜑 → (((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (0_g‘𝐾) ↔ 𝐵 = 𝐶))
150	149	bicomd 222	. . 3 ⊢ (𝜑 → (𝐵 = 𝐶 ↔ ((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (0_g‘𝐾)))
151	144	zrhrhm 21444	. . . . . . 7 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
152		rhmghm 20430	. . . . . . 7 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
153	151, 152	syl 17	. . . . . 6 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
154	142, 153	syl 17	. . . . 5 ⊢ (𝜑 → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
155		0zd 12608	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
156	155, 7	zsubcld 12709	. . . . . 6 ⊢ (𝜑 → (0 − 𝐶) ∈ ℤ)
157		zringbas 21386	. . . . . 6 ⊢ ℤ = (Base‘ℤ_ring)
158	156, 157	eleqtrdi 2839	. . . . 5 ⊢ (𝜑 → (0 − 𝐶) ∈ (Base‘ℤ_ring))
159	10, 157	eleqtrdi 2839	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Base‘ℤ_ring))
160		eqid 2728	. . . . . 6 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
161		eqid 2728	. . . . . 6 ⊢ (+_g‘ℤ_ring) = (+_g‘ℤ_ring)
162		eqid 2728	. . . . . 6 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
163	160, 161, 162	ghmlin 19182	. . . . 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 1368	. . . 4 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝐶)(+_g‘ℤ_ring)𝐵)) = (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)))
165	164	eqeq1d 2730	. . 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 278	. 2 ⊢ (𝜑 → (𝐵 = 𝐶 ↔ (((ℤRHom‘𝐾)‘(0 − 𝐶))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝐵)) = (0_g‘𝐾)))
167		eqid 2728	. . . . . 6 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
168		eqid 2728	. . . . . 6 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
169		eqid 2728	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
170		eqid 2728	. . . . . 6 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
171	157, 169	ghmf 19181	. . . . . . . 8 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
172	154, 171	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
173	172, 156	ffvelcdmd 7100	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘(0 − 𝐶)) ∈ (Base‘𝐾))
174		aks6d1c5.6	. . . . . . 7 ⊢ 𝑋 = (var₁‘𝐾)
175	167, 174, 169, 168, 170, 140, 173	evl1vard 22263	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = ((ℤRHom‘𝐾)‘(0 − 𝐶))))
176		eqid 2728	. . . . . . 7 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
177	172, 10	ffvelcdmd 7100	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝐵) ∈ (Base‘𝐾))
178	167, 168, 169, 176, 170, 140, 177, 173	evl1scad 22261	. . . . . 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 2728	. . . . . 6 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
180	167, 168, 169, 170, 140, 173, 175, 178, 179, 162	evl1addd 22267	. . . . 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 494	. . . 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 2734	. . 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 2730	. 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 278	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ↦ cmpt 5235 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ↑_m cmap 8851 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 < clt 11286 ≤ cle 11287 − cmin 11482 -cneg 11483 ℕcn 12250 ℕ₀cn0 12510 ℤcz 12596 ...cfz 13524 ∥ cdvds 16238 ℙcprime 16649 Basecbs 17187 +_gcplusg 17240 0_gc0g 17428 Σ_g cgsu 17429 ._gcmg 19030 GrpHom cghm 19174 mulGrpcmgp 20081 Ringcrg 20180 CRingccrg 20181 RingHom crh 20415 Fieldcfield 20632 ℤ_ringczring 21379 ℤRHomczrh 21432 chrcchr 21434 algSccascl 21793 var₁cv1 22102 Poly₁cpl1 22103 eval₁ce1 22240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-dvds 16239 df-prm 16650 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-od 19490 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-field 20634 df-lmod 20752 df-lss 20823 df-lsp 20863 df-cnfld 21287 df-zring 21380 df-zrh 21436 df-chr 21438 df-assa 21794 df-asp 21795 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-opsr 21853 df-evls 22025 df-evl 22026 df-psr1 22106 df-vr1 22107 df-ply1 22108 df-evl1 22242

This theorem is referenced by: aks6d1c5lem2 41641

W3C validator