Theorem aks6d1c5 42172

Description: Claim 5 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. The mapping defined by 𝐺 is injective. (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)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))

Assertion

Ref	Expression
aks6d1c5	⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))–1-1→(Base‘(Poly1‘𝐾)))

Distinct variable groups:   𝐴,𝑔,𝑖   𝑔,𝐾,𝑖   𝜑,𝑔,𝑖   ↑ ,𝑔,𝑖   𝑔,𝐺,𝑖   𝑔,𝑋,𝑖

Allowed substitution hints:   𝑃(𝑔,𝑖)

Proof of Theorem aks6d1c5

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . . 6 ⊢ (Base‘(mulGrp‘(Poly1‘𝐾))) = (Base‘(mulGrp‘(Poly1‘𝐾)))
2		aks6d1p5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Field)
3	2	fldcrngd 20652	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CRing)
4		eqid 2731	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
5	4	ply1crng 22106	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → (Poly1‘𝐾) ∈ CRing)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CRing)
7		eqid 2731	. . . . . . . . 9 ⊢ (mulGrp‘(Poly1‘𝐾)) = (mulGrp‘(Poly1‘𝐾))
8	7	crngmgp 20154	. . . . . . . 8 ⊢ ((Poly1‘𝐾) ∈ CRing → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
9	6, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
11		fzfid 13875	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (0...𝐴) ∈ Fin)
12		aks6d1c5.7	. . . . . . . 8 ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))
13	10	cmnmndd 19711	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (mulGrp‘(Poly1‘𝐾)) ∈ Mnd)
14	13	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (mulGrp‘(Poly1‘𝐾)) ∈ Mnd)
15		nn0ex 12382	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ0 ∈ V)
17		ovexd 7376	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝐴) ∈ V)
18	16, 17	elmapd 8759	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑔:(0...𝐴)⟶ℕ0))
19	18	biimpd 229	. . . . . . . . . 10 ⊢ (𝜑 → (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) → 𝑔:(0...𝐴)⟶ℕ0))
20	19	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → 𝑔:(0...𝐴)⟶ℕ0)
21	20	ffvelcdmda 7012	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) ∈ ℕ0)
22	6	crngringd 20159	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Ring)
23	22	ringcmnd 20197	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CMnd)
24		cmnmnd 19704	. . . . . . . . . . . . 13 ⊢ ((Poly1‘𝐾) ∈ CMnd → (Poly1‘𝐾) ∈ Mnd)
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Mnd)
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (Poly1‘𝐾) ∈ Mnd)
27	26	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (Poly1‘𝐾) ∈ Mnd)
28	3	crngringd 20159	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → 𝐾 ∈ Ring)
30	29	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝐾 ∈ Ring)
31		aks6d1c5.6	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝐾)
32		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
33	31, 4, 32	vr1cl 22125	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘(Poly1‘𝐾)))
34	30, 33	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝑋 ∈ (Base‘(Poly1‘𝐾)))
35		simpl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))))
36		elfzelz 13419	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
37	36	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ ℤ)
38	35, 37	jca 511	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ ℤ))
39		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
40	39	zrhrhm 21443	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
41		zringbas 21385	. . . . . . . . . . . . . . . 16 ⊢ ℤ = (Base‘ℤring)
42		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
43	41, 42	rhmf 20397	. . . . . . . . . . . . . . 15 ⊢ ((ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
44	40, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
45	29, 44	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
46	45	ffvelcdmda 7012	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ ℤ) → ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾))
47	38, 46	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾))
48		eqid 2731	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾))
49	4, 48, 42, 32	ply1sclcl 22195	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1‘𝐾)))
50	30, 47, 49	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1‘𝐾)))
51		eqid 2731	. . . . . . . . . . 11 ⊢ (+g‘(Poly1‘𝐾)) = (+g‘(Poly1‘𝐾))
52	32, 51	mndcl 18645	. . . . . . . . . 10 ⊢ (((Poly1‘𝐾) ∈ Mnd ∧ 𝑋 ∈ (Base‘(Poly1‘𝐾)) ∧ ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1‘𝐾))) → (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly1‘𝐾)))
53	27, 34, 50, 52	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly1‘𝐾)))
54	7, 32	mgpbas 20058	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(mulGrp‘(Poly1‘𝐾)))
55	54	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (Base‘(Poly1‘𝐾)) = (Base‘(mulGrp‘(Poly1‘𝐾))))
56	53, 55	eleqtrd 2833	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(mulGrp‘(Poly1‘𝐾))))
57	1, 12, 14, 21, 56	mulgnn0cld 19003	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(mulGrp‘(Poly1‘𝐾))))
58	57	ralrimiva 3124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → ∀𝑖 ∈ (0...𝐴)((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(mulGrp‘(Poly1‘𝐾))))
59	1, 10, 11, 58	gsummptcl 19874	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(mulGrp‘(Poly1‘𝐾))))
60	54	eqcomi 2740	. . . . . 6 ⊢ (Base‘(mulGrp‘(Poly1‘𝐾))) = (Base‘(Poly1‘𝐾))
61	60	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (Base‘(mulGrp‘(Poly1‘𝐾))) = (Base‘(Poly1‘𝐾)))
62	59, 61	eleqtrd 2833	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(Poly1‘𝐾)))
63		aks6d1c5.8	. . . 4 ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
64	62, 63	fmptd 7042	. . 3 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)))
65		eqidd 2732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (0g‘𝐾) = (0g‘𝐾))
66		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦)
67	66	neneqd 2933	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ¬ 𝑥 = 𝑦)
68		simp-4r 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ (ℕ0 ↑m (0...𝐴)))
69	15	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ℕ0 ∈ V)
70		ovexd 7376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (0...𝐴) ∈ V)
71	69, 70	elmapd 8759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (𝑥 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑥:(0...𝐴)⟶ℕ0))
72	68, 71	mpbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥:(0...𝐴)⟶ℕ0)
73		ffn 6646	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥:(0...𝐴)⟶ℕ0 → 𝑥 Fn (0...𝐴))
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥 Fn (0...𝐴))
75		simpllr 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ (ℕ0 ↑m (0...𝐴)))
76	69, 70	elmapd 8759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (𝑦 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑦:(0...𝐴)⟶ℕ0))
77	75, 76	mpbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑦:(0...𝐴)⟶ℕ0)
78		ffn 6646	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:(0...𝐴)⟶ℕ0 → 𝑦 Fn (0...𝐴))
79	77, 78	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑦 Fn (0...𝐴))
80		eqfnfv2 6960	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 Fn (0...𝐴) ∧ 𝑦 Fn (0...𝐴)) → (𝑥 = 𝑦 ↔ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧))))
81	74, 79, 80	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (𝑥 = 𝑦 ↔ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧))))
82	81	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (¬ 𝑥 = 𝑦 ↔ ¬ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧))))
83	82	biimpd 229	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (¬ 𝑥 = 𝑦 → ¬ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧))))
84	67, 83	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ¬ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧)))
85		ianor 983	. . . . . . . . . . . . . . 15 ⊢ (¬ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧)) ↔ (¬ (0...𝐴) = (0...𝐴) ∨ ¬ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧)))
86	84, 85	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (¬ (0...𝐴) = (0...𝐴) ∨ ¬ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧)))
87		eqidd 2732	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (0...𝐴) = (0...𝐴))
88	87	notnotd 144	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ¬ ¬ (0...𝐴) = (0...𝐴))
89	86, 88	orcnd 878	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ¬ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧))
90		rexnal 3084	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ (0...𝐴) ¬ (𝑥‘𝑧) = (𝑦‘𝑧) ↔ ¬ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧))
91	89, 90	sylibr 234	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ∃𝑧 ∈ (0...𝐴) ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
92		df-ne 2929	. . . . . . . . . . . . 13 ⊢ ((𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
93	92	rexbii 3079	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (0...𝐴)(𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ∃𝑧 ∈ (0...𝐴) ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
94	91, 93	sylibr 234	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ∃𝑧 ∈ (0...𝐴)(𝑥‘𝑧) ≠ (𝑦‘𝑧))
95		simpl 482	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥‘𝑧) ≠ (𝑦‘𝑧))) → ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦))
96		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥‘𝑧) ≠ (𝑦‘𝑧))) → 𝑧 ∈ (0...𝐴))
97		simprr 772	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥‘𝑧) ≠ (𝑦‘𝑧))) → (𝑥‘𝑧) ≠ (𝑦‘𝑧))
98	95, 96, 97	jca31 514	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥‘𝑧) ≠ (𝑦‘𝑧))) → ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) ≠ (𝑦‘𝑧)))
99	71	biimpd 229	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (𝑥 ∈ (ℕ0 ↑m (0...𝐴)) → 𝑥:(0...𝐴)⟶ℕ0))
100	68, 99	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥:(0...𝐴)⟶ℕ0)
101	100	ffvelcdmda 7012	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑥‘𝑧) ∈ ℕ0)
102	101	nn0red 12438	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑥‘𝑧) ∈ ℝ)
103	76	biimpd 229	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (𝑦 ∈ (ℕ0 ↑m (0...𝐴)) → 𝑦:(0...𝐴)⟶ℕ0))
104	75, 103	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑦:(0...𝐴)⟶ℕ0)
105	104	ffvelcdmda 7012	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑦‘𝑧) ∈ ℕ0)
106	105	nn0red 12438	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑦‘𝑧) ∈ ℝ)
107	102, 106	lttri2d 11247	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) → ((𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ((𝑥‘𝑧) < (𝑦‘𝑧) ∨ (𝑦‘𝑧) < (𝑥‘𝑧))))
108	2	ad6antr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → 𝐾 ∈ Field)
109		aks6d1p5.2	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℙ)
110	109	ad6antr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → 𝑃 ∈ ℙ)
111		aks6d1c5.3	. . . . . . . . . . . . . . . . 17 ⊢ 𝑃 = (chr‘𝐾)
112		aks6d1c5.4	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
113	112	ad6antr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → 𝐴 ∈ ℕ0)
114		aks6d1c5.5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 < 𝑃)
115	114	ad6antr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → 𝐴 < 𝑃)
116	68	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → 𝑥 ∈ (ℕ0 ↑m (0...𝐴)))
117	75	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → 𝑦 ∈ (ℕ0 ↑m (0...𝐴)))
118		simp-4r 783	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → (𝐺‘𝑥) = (𝐺‘𝑦))
119		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → 𝑧 ∈ (0...𝐴))
120		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → (𝑥‘𝑧) < (𝑦‘𝑧))
121	108, 110, 111, 113, 115, 31, 12, 63, 116, 117, 118, 119, 120	aks6d1c5lem2 42171	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) < (𝑦‘𝑧)) → (0g‘𝐾) ≠ (0g‘𝐾))
122	2	ad6antr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → 𝐾 ∈ Field)
123	109	ad6antr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → 𝑃 ∈ ℙ)
124	112	ad6antr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → 𝐴 ∈ ℕ0)
125	114	ad6antr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → 𝐴 < 𝑃)
126	75	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → 𝑦 ∈ (ℕ0 ↑m (0...𝐴)))
127	68	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → 𝑥 ∈ (ℕ0 ↑m (0...𝐴)))
128		simp-4r 783	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → (𝐺‘𝑥) = (𝐺‘𝑦))
129	128	eqcomd 2737	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → (𝐺‘𝑦) = (𝐺‘𝑥))
130		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → 𝑧 ∈ (0...𝐴))
131		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → (𝑦‘𝑧) < (𝑥‘𝑧))
132	122, 123, 111, 124, 125, 31, 12, 63, 126, 127, 129, 130, 131	aks6d1c5lem2 42171	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦‘𝑧) < (𝑥‘𝑧)) → (0g‘𝐾) ≠ (0g‘𝐾))
133	121, 132	jaodan 959	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ ((𝑥‘𝑧) < (𝑦‘𝑧) ∨ (𝑦‘𝑧) < (𝑥‘𝑧))) → (0g‘𝐾) ≠ (0g‘𝐾))
134	133	ex 412	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (((𝑥‘𝑧) < (𝑦‘𝑧) ∨ (𝑦‘𝑧) < (𝑥‘𝑧)) → (0g‘𝐾) ≠ (0g‘𝐾)))
135	107, 134	sylbid 240	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) → ((𝑥‘𝑧) ≠ (𝑦‘𝑧) → (0g‘𝐾) ≠ (0g‘𝐾)))
136	135	imp 406	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥‘𝑧) ≠ (𝑦‘𝑧)) → (0g‘𝐾) ≠ (0g‘𝐾))
137	98, 136	syl 17	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥‘𝑧) ≠ (𝑦‘𝑧))) → (0g‘𝐾) ≠ (0g‘𝐾))
138	94, 137	rexlimddv 3139	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (0g‘𝐾) ≠ (0g‘𝐾))
139	138	neneqd 2933	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ¬ (0g‘𝐾) = (0g‘𝐾))
140	65, 139	pm2.65da 816	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → ¬ 𝑥 ≠ 𝑦)
141		df-ne 2929	. . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
142	141	notbii 320	. . . . . . . 8 ⊢ (¬ 𝑥 ≠ 𝑦 ↔ ¬ ¬ 𝑥 = 𝑦)
143	140, 142	sylib 218	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → ¬ ¬ 𝑥 = 𝑦)
144		notnotb 315	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ ¬ ¬ 𝑥 = 𝑦)
145	143, 144	sylibr 234	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → 𝑥 = 𝑦)
146	145	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) → ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
147	146	ralrimiva 3124	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) → ∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
148	147	ralrimiva 3124	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (ℕ0 ↑m (0...𝐴))∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
149	64, 148	jca 511	. 2 ⊢ (𝜑 → (𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)) ∧ ∀𝑥 ∈ (ℕ0 ↑m (0...𝐴))∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦)))
150		dff13 7183	. 2 ⊢ (𝐺:(ℕ0 ↑m (0...𝐴))–1-1→(Base‘(Poly1‘𝐾)) ↔ (𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)) ∧ ∀𝑥 ∈ (ℕ0 ↑m (0...𝐴))∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦)))
151	149, 150	sylibr 234	1 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))–1-1→(Base‘(Poly1‘𝐾)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  Vcvv 3436   class class class wbr 5086   ↦ cmpt 5167   Fn wfn 6471  ⟶wf 6472  –1-1→wf1 6473  ‘cfv 6476  (class class class)co 7341   ↑_m cmap 8745  0cc0 11001   < clt 11141  ℕ₀cn0 12376  ℤcz 12463  ...cfz 13402  ℙcprime 16577  Basecbs 17115  +_gcplusg 17156  0_gc0g 17338   Σ_g cgsu 17339  Mndcmnd 18637  ._gcmg 18975  CMndccmn 19687  mulGrpcmgp 20053  Ringcrg 20146  CRingccrg 20147   RingHom crh 20382  Fieldcfield 20640  ℤ_ringczring 21378  ℤRHomczrh 21431  chrcchr 21433  algSccascl 21784  var₁cv1 22083  Poly₁cpl1 22084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078  ax-pre-sup 11079  ax-addf 11080  ax-mulf 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-sup 9321  df-inf 9322  df-oi 9391  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-div 11770  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-uz 12728  df-rp 12886  df-fz 13403  df-fzo 13550  df-fl 13691  df-mod 13769  df-seq 13904  df-exp 13964  df-hash 14233  df-cj 15001  df-re 15002  df-im 15003  df-sqrt 15137  df-abs 15138  df-dvds 16159  df-prm 16578  df-struct 17053  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-mulr 17170  df-starv 17171  df-sca 17172  df-vsca 17173  df-ip 17174  df-tset 17175  df-ple 17176  df-ds 17178  df-unif 17179  df-hom 17180  df-cco 17181  df-0g 17340  df-gsum 17341  df-prds 17346  df-pws 17348  df-mre 17483  df-mrc 17484  df-acs 17486  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-ghm 19120  df-cntz 19224  df-od 19435  df-cmn 19689  df-abl 19690  df-mgp 20054  df-rng 20066  df-ur 20095  df-srg 20100  df-ring 20148  df-cring 20149  df-oppr 20250  df-dvdsr 20270  df-unit 20271  df-invr 20301  df-rhm 20385  df-nzr 20423  df-subrng 20456  df-subrg 20480  df-rlreg 20604  df-domn 20605  df-idom 20606  df-drng 20641  df-field 20642  df-lmod 20790  df-lss 20860  df-lsp 20900  df-cnfld 21287  df-zring 21379  df-zrh 21435  df-chr 21437  df-assa 21785  df-asp 21786  df-ascl 21787  df-psr 21841  df-mvr 21842  df-mpl 21843  df-opsr 21845  df-evls 22004  df-evl 22005  df-psr1 22087  df-vr1 22088  df-ply1 22089  df-coe1 22090  df-evl1 22226  df-mdeg 25982  df-deg1 25983  df-uc1p 26059  df-q1p 26060

This theorem is referenced by:  aks6d1c6lem3  42205