Theorem aks6d1c5 41642

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 2728	. . . . . 6 ⊢ (Base‘(mulGrp‘(Poly1‘𝐾))) = (Base‘(mulGrp‘(Poly1‘𝐾)))
2		aks6d1p5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Field)
3	2	fldcrngd 20644	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CRing)
4		eqid 2728	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
5	4	ply1crng 22124	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → (Poly1‘𝐾) ∈ CRing)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CRing)
7		eqid 2728	. . . . . . . . 9 ⊢ (mulGrp‘(Poly1‘𝐾)) = (mulGrp‘(Poly1‘𝐾))
8	7	crngmgp 20188	. . . . . . . 8 ⊢ ((Poly1‘𝐾) ∈ CRing → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
9	6, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
10	9	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (mulGrp‘(Poly1‘𝐾)) ∈ CMnd)
11		fzfid 13978	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (0...𝐴) ∈ Fin)
12		aks6d1c5.7	. . . . . . . 8 ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))
13	10	cmnmndd 19766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (mulGrp‘(Poly1‘𝐾)) ∈ Mnd)
14	13	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (mulGrp‘(Poly1‘𝐾)) ∈ Mnd)
15		nn0ex 12516	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ0 ∈ V)
17		ovexd 7461	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝐴) ∈ V)
18	16, 17	elmapd 8865	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑔:(0...𝐴)⟶ℕ0))
19	18	biimpd 228	. . . . . . . . . 10 ⊢ (𝜑 → (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) → 𝑔:(0...𝐴)⟶ℕ0))
20	19	imp 405	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → 𝑔:(0...𝐴)⟶ℕ0)
21	20	ffvelcdmda 7099	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) ∈ ℕ0)
22	6	crngringd 20193	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Ring)
23	22	ringcmnd 20227	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CMnd)
24		cmnmnd 19759	. . . . . . . . . . . . 13 ⊢ ((Poly1‘𝐾) ∈ CMnd → (Poly1‘𝐾) ∈ Mnd)
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Mnd)
26	25	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (Poly1‘𝐾) ∈ Mnd)
27	26	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (Poly1‘𝐾) ∈ Mnd)
28	3	crngringd 20193	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
29	28	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → 𝐾 ∈ Ring)
30	29	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝐾 ∈ Ring)
31		aks6d1c5.6	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝐾)
32		eqid 2728	. . . . . . . . . . . 12 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
33	31, 4, 32	vr1cl 22143	. . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))))
36		elfzelz 13541	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
37	36	adantl 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ ℤ)
38	35, 37	jca 510	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ ℤ))
39		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
40	39	zrhrhm 21444	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
41		zringbas 21386	. . . . . . . . . . . . . . . 16 ⊢ ℤ = (Base‘ℤring)
42		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
43	41, 42	rhmf 20431	. . . . . . . . . . . . . . 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 7099	. . . . . . . . . . . 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 2728	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾))
49	4, 48, 42, 32	ply1sclcl 22212	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1‘𝐾)))
50	30, 47, 49	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1‘𝐾)))
51		eqid 2728	. . . . . . . . . . 11 ⊢ (+g‘(Poly1‘𝐾)) = (+g‘(Poly1‘𝐾))
52	32, 51	mndcl 18709	. . . . . . . . . 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 1368	. . . . . . . . 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 20087	. . . . . . . . . 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 2831	. . . . . . . 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 19057	. . . . . . 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 3143	. . . . . 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 19929	. . . . 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 2737	. . . . . 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 2831	. . . 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 7129	. . 3 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)))
65		eqidd 2729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → (0g‘𝐾) = (0g‘𝐾))
66		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦)
67	66	neneqd 2942	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → ¬ 𝑥 = 𝑦)
68		simp-4r 782	. . . . . . . . . . . . . . . . . . . . 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 7461	. . . . . . . . . . . . . . . . . . . . . 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 8865	. . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑥:(0...𝐴)⟶ℕ0)
73		ffn 6727	. . . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . 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 8865	. . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ∧ 𝑥 ≠ 𝑦) → 𝑦:(0...𝐴)⟶ℕ0)
78		ffn 6727	. . . . . . . . . . . . . . . . . . . 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 7046	. . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . 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 317	. . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . 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 979	. . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . 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 876	. . . . . . . . . . . . 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 3097	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ (0...𝐴) ¬ (𝑥‘𝑧) = (𝑦‘𝑧) ↔ ¬ ∀𝑧 ∈ (0...𝐴)(𝑥‘𝑧) = (𝑦‘𝑧))
91	89, 90	sylibr 233	. . . . . . . . . . . 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 2938	. . . . . . . . . . . . 13 ⊢ ((𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
93	92	rexbii 3091	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (0...𝐴)(𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ∃𝑧 ∈ (0...𝐴) ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
94	91, 93	sylibr 233	. . . . . . . . . . 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 481	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 513	. . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . 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 7099	. . . . . . . . . . . . . . . 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 12571	. . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . 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 7099	. . . . . . . . . . . . . . . 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 12571	. . . . . . . . . . . . . . 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 11391	. . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 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 41641	. . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 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 41641	. . . . . . . . . . . . . . . 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 955	. . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . 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 239	. . . . . . . . . . . . 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 405	. . . . . . . . . . . 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 3158	. . . . . . . . . 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 2942	. . . . . . . . 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 815	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → ¬ 𝑥 ≠ 𝑦)
141		df-ne 2938	. . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
142	141	notbii 319	. . . . . . . 8 ⊢ (¬ 𝑥 ≠ 𝑦 ↔ ¬ ¬ 𝑥 = 𝑦)
143	140, 142	sylib 217	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → ¬ ¬ 𝑥 = 𝑦)
144		notnotb 314	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ ¬ ¬ 𝑥 = 𝑦)
145	143, 144	sylibr 233	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) → 𝑥 = 𝑦)
146	145	ex 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0 ↑m (0...𝐴))) → ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
147	146	ralrimiva 3143	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) → ∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
148	147	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (ℕ0 ↑m (0...𝐴))∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
149	64, 148	jca 510	. 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 7271	. 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 233	1 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))–1-1→(Base‘(Poly1‘𝐾)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3473   class class class wbr 5152   ↦ cmpt 5235   Fn wfn 6548  ⟶wf 6549  –1-1→wf1 6550  ‘cfv 6553  (class class class)co 7426   ↑_m cmap 8851  0cc0 11146   < clt 11286  ℕ₀cn0 12510  ℤcz 12596  ...cfz 13524  ℙcprime 16649  Basecbs 17187  +_gcplusg 17240  0_gc0g 17428   Σ_g cgsu 17429  Mndcmnd 18701  ._gcmg 19030  CMndccmn 19742  mulGrpcmgp 20081  Ringcrg 20180  CRingccrg 20181   RingHom crh 20415  Fieldcfield 20632  ℤ_ringczring 21379  ℤRHomczrh 21432  chrcchr 21434  algSccascl 21793  var₁cv1 22102  Poly₁cpl1 22103

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-tpos 8238  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-oppr 20280  df-dvdsr 20303  df-unit 20304  df-invr 20334  df-rhm 20418  df-nzr 20459  df-subrng 20490  df-subrg 20515  df-drng 20633  df-field 20634  df-lmod 20752  df-lss 20823  df-lsp 20863  df-rlreg 21237  df-domn 21238  df-idom 21239  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-coe1 22109  df-evl1 22242  df-mdeg 26008  df-deg1 26009  df-uc1p 26087  df-q1p 26088

This theorem is referenced by:  aks6d1c6lem3  41676