Theorem aks6d1c5 43455

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 2763	. . . . . 6 ⊢ (Base‘(mulGrp‘(Poly1‘𝐾))) = (Base‘(mulGrp‘(Poly1‘𝐾)))
2		aks6d1p5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Field)
3	2	fldcrngd 21046	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CRing)
4		eqid 2763	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
5	4	ply1crng 22571	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → (Poly1‘𝐾) ∈ CRing)
6	3, 5	syl 18	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CRing)
7		eqid 2763	. . . . . . . . 9 ⊢ (mulGrp‘(Poly1‘𝐾)) = (mulGrp‘(Poly1‘𝐾))
8	7	crngmgp 20518	. . . . . . . 8 ⊢ ((Poly1‘𝐾) ∈ CRing → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
9	6, 8	syl 18	. . . . . . 7 ⊢ (𝜑 → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
11		fzfid 14156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (0...𝐴) ∈ Fin)
12		aks6d1c5.7	. . . . . . . 8 ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))
13	10	cmnmndd 20069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (mulGrp‘(Poly1‘𝐾)) ∈ Mnd)
14	13	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (mulGrp‘(Poly1‘𝐾)) ∈ Mnd)
15		nn0ex 12652	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ0 ∈ V)
17		ovexd 7486	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝐴) ∈ V)
18	16, 17	elmapd 8898	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑔:(0...𝐴)⟶ℕ0))
19	18	biimpd 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) → 𝑔:(0...𝐴)⟶ℕ0))
20	19	imp 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → 𝑔:(0...𝐴)⟶ℕ0)
21	20	ffvelcdmda 7116	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) ∈ ℕ0)
22	6	crngringd 20524	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Ring)
23	22	ringcmnd 20564	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CMnd)
24		cmnmnd 20062	. . . . . . . . . . . . 13 ⊢ ((Poly1‘𝐾) ∈ CMnd → (Poly1‘𝐾) ∈ Mnd)
25	23, 24	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Mnd)
26	25	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (Poly1‘𝐾) ∈ Mnd)
27	26	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (Poly1‘𝐾) ∈ Mnd)
28	3	crngringd 20524	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
29	28	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → 𝐾 ∈ Ring)
30	29	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝐾 ∈ Ring)
31		aks6d1c5.6	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝐾)
32		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
33	31, 4, 32	vr1cl 22590	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘(Poly1‘𝐾)))
34	30, 33	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝑋 ∈ (Base‘(Poly1‘𝐾)))
35		simpl 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))))
36		elfzelz 13696	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
37	36	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ ℤ)
38	35, 37	jca 521	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ ℤ))
39		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
40	39	zrhrhm 21872	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
41		zringbas 21813	. . . . . . . . . . . . . . . 16 ⊢ ℤ = (Base‘ℤring)
42		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
43	41, 42	rhmf 20766	. . . . . . . . . . . . . . 15 ⊢ ((ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
44	40, 43	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
45	29, 44	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
46	45	ffvelcdmda 7116	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ ℤ) → ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾))
47	38, 46	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾))
48		eqid 2763	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾))
49	4, 48, 42, 32	ply1sclcl 22660	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1‘𝐾)))
50	30, 47, 49	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1‘𝐾)))
51		eqid 2763	. . . . . . . . . . 11 ⊢ (+g‘(Poly1‘𝐾)) = (+g‘(Poly1‘𝐾))
52	32, 51	mndcl 18981	. . . . . . . . . 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 1399	. . . . . . . . 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 20416	. . . . . . . . . 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 2866	. . . . . . . 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 19355	. . . . . . 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 3158	. . . . . 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 20232	. . . . 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 2772	. . . . . 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 2866	. . . 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 7146	. . 3 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)))
65		eqidd 2764	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (0g‘𝐾) = (0g‘𝐾))
66		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦)
67	66	neneqd 2964	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ¬ 𝑥 = 𝑦)
68		simp-4r 796	. . . . . . . . . . . . . . . . . . . . 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 7486	. . . . . . . . . . . . . . . . . . . . . 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 8898	. . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥:(0...𝐴)⟶ℕ0)
73		ffn 6739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥:(0...𝐴)⟶ℕ0 → 𝑥 Fn (0...𝐴))
74	72, 73	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥 Fn (0...𝐴))
75		simpllr 788	. . . . . . . . . . . . . . . . . . . . 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 8898	. . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑦:(0...𝐴)⟶ℕ0)
78		ffn 6739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:(0...𝐴)⟶ℕ0 → 𝑦 Fn (0...𝐴))
79	77, 78	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑦 Fn (0...𝐴))
80		eqfnfv2 7061	. . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . 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 321	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 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 997	. . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 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 145	. . . . . . . . . . . . . 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 892	. . . . . . . . . . . . 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 3118	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ (0...𝐴) ¬ (𝑥‘𝑧) = (𝑦‘𝑧) ↔ ¬ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧))
91	89, 90	sylibr 237	. . . . . . . . . . . 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 2960	. . . . . . . . . . . . 13 ⊢ ((𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
93	92	rexbii 3113	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (0...𝐴)(𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ∃𝑧 ∈ (0...𝐴) ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
94	91, 93	sylibr 237	. . . . . . . . . . 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 488	. . . . . . . . . . . . 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 783	. . . . . . . . . . . . 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 785	. . . . . . . . . . . . 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 524	. . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . 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 7116	. . . . . . . . . . . . . . . 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 12708	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . 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 7116	. . . . . . . . . . . . . . . 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 12708	. . . . . . . . . . . . . . 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 11488	. . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . 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 43454	. . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . 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 43454	. . . . . . . . . . . . . . . 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 972	. . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . 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 243	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 3173	. . . . . . . . . 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 2964	. . . . . . . . 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 829	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → ¬ 𝑥 ≠ 𝑦)
141		df-ne 2960	. . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
142	141	notbii 323	. . . . . . . 8 ⊢ (¬ 𝑥 ≠ 𝑦 ↔ ¬ ¬ 𝑥 = 𝑦)
143	140, 142	sylib 221	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → ¬ ¬ 𝑥 = 𝑦)
144		notnotb 318	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ ¬ ¬ 𝑥 = 𝑦)
145	143, 144	sylibr 237	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → 𝑥 = 𝑦)
146	145	ex 418	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) → ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
147	146	ralrimiva 3158	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) → ∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
148	147	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (ℕ0 ↑m (0...𝐴))∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
149	64, 148	jca 521	. 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 7289	. 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 237	1 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))⟶1-1(Base‘(Poly1‘𝐾)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   class class class wbr 5108   ↦ cmpt 5192   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  0cc0 11239   < clt 11382  ℕ₀cn0 12646  ℤcz 12733  ...cfz 13679  ℙcprime 16890  Basecbs 17431  +_gcplusg 17472  0_gc0g 17660   Σ_g cgsu 17661  Mndcmnd 18973  ._gcmg 19327  CMndccmn 20045  mulGrpcmgp 20411  Ringcrg 20510  CRingccrg 20511   RingHom crh 20750  Fieldcfield 21032  ℤ_ringczring 21806  ℤRHomczrh 21860  chrcchr 21862  algSccascl 22215  var₁cv1 22549  Poly₁cpl1 22550

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317  ax-addf 11318  ax-mulf 11319

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-ofr 7726  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-pm 8888  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-sup 9471  df-inf 9472  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-fl 13972  df-mod 14050  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-dvds 16467  df-prm 16891  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-starv 17488  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-unif 17496  df-hom 17497  df-cco 17498  df-euf 17499  df-0g 17662  df-gsum 17663  df-prds 17668  df-pws 17670  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-ghm 19478  df-cntz 19582  df-od 19793  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-srg 20464  df-ring 20512  df-cring 20513  df-oppr 20618  df-dvdsr 20638  df-unit 20639  df-invr 20669  df-rhm 20753  df-nzr 20814  df-subrng 20849  df-subrg 20873  df-rlreg 20997  df-domn 20998  df-idom 20999  df-drng 21033  df-field 21034  df-lmod 21188  df-lss 21258  df-lsp 21298  df-cnfld 21730  df-zring 21807  df-zrh 21864  df-chr 21866  df-assa 22216  df-asp 22217  df-ascl 22218  df-psr 22272  df-mvr 22273  df-mpl 22274  df-opsr 22276  df-evls 22438  df-evl 22439  df-psr1 22553  df-vr1 22554  df-ply1 22555  df-coe1 22556  df-evl1 22689  df-mdeg 26428  df-deg1 26429  df-uc1p 26505  df-q1p 26506

This theorem is used by:  aks6d1c6lem3  43488