Theorem aks6d1c5 42134

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 2730	. . . . . 6 ⊢ (Base‘(mulGrp‘(Poly1‘𝐾))) = (Base‘(mulGrp‘(Poly1‘𝐾)))
2		aks6d1p5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Field)
3	2	fldcrngd 20658	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CRing)
4		eqid 2730	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
5	4	ply1crng 22090	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → (Poly1‘𝐾) ∈ CRing)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CRing)
7		eqid 2730	. . . . . . . . 9 ⊢ (mulGrp‘(Poly1‘𝐾)) = (mulGrp‘(Poly1‘𝐾))
8	7	crngmgp 20157	. . . . . . . 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 13945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (0...𝐴) ∈ Fin)
12		aks6d1c5.7	. . . . . . . 8 ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))
13	10	cmnmndd 19741	. . . . . . . . 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 12455	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ0 ∈ V)
17		ovexd 7425	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝐴) ∈ V)
18	16, 17	elmapd 8816	. . . . . . . . . . 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 7059	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) ∈ ℕ0)
22	6	crngringd 20162	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Ring)
23	22	ringcmnd 20200	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CMnd)
24		cmnmnd 19734	. . . . . . . . . . . . 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 20162	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
33	31, 4, 32	vr1cl 22109	. . . . . . . . . . 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 13492	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . 16 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
40	39	zrhrhm 21428	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
41		zringbas 21370	. . . . . . . . . . . . . . . 16 ⊢ ℤ = (Base‘ℤring)
42		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
43	41, 42	rhmf 20401	. . . . . . . . . . . . . . 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 7059	. . . . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾))
49	4, 48, 42, 32	ply1sclcl 22179	. . . . . . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (+g‘(Poly1‘𝐾)) = (+g‘(Poly1‘𝐾))
52	32, 51	mndcl 18676	. . . . . . . . . 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 20061	. . . . . . . . . 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 19034	. . . . . . 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 3126	. . . . . 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 19904	. . . . 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 2739	. . . . . 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 7089	. . 3 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)))
65		eqidd 2731	. . . . . . . . 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 2931	. . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . . . . . . . . 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 8816	. . . . . . . . . . . . . . . . . . . . 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 6691	. . . . . . . . . . . . . . . . . . . 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 8816	. . . . . . . . . . . . . . . . . . . . 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 6691	. . . . . . . . . . . . . . . . . . . 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 7007	. . . . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . 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 3083	. . . . . . . . . . . . 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 2927	. . . . . . . . . . . . 13 ⊢ ((𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
93	92	rexbii 3077	. . . . . . . . . . . 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 7059	. . . . . . . . . . . . . . . 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 12511	. . . . . . . . . . . . . . 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 7059	. . . . . . . . . . . . . . . 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 12511	. . . . . . . . . . . . . . 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 11320	. . . . . . . . . . . . . 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 42133	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 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 42133	. . . . . . . . . . . . . . . 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 3141	. . . . . . . . . 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 2931	. . . . . . . . 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 2927	. . . . . . . . 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 3126	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) → ∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
148	147	ralrimiva 3126	. . 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 7232	. 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 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3450   class class class wbr 5110   ↦ cmpt 5191   Fn wfn 6509  ⟶wf 6510  –1-1→wf1 6511  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  0cc0 11075   < clt 11215  ℕ₀cn0 12449  ℤcz 12536  ...cfz 13475  ℙcprime 16648  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409   Σ_g cgsu 17410  Mndcmnd 18668  ._gcmg 19006  CMndccmn 19717  mulGrpcmgp 20056  Ringcrg 20149  CRingccrg 20150   RingHom crh 20385  Fieldcfield 20646  ℤ_ringczring 21363  ℤRHomczrh 21416  chrcchr 21418  algSccascl 21768  var₁cv1 22067  Poly₁cpl1 22068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154  ax-mulf 11155

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-fl 13761  df-mod 13839  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-dvds 16230  df-prm 16649  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-cntz 19256  df-od 19465  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-srg 20103  df-ring 20151  df-cring 20152  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-rhm 20388  df-nzr 20429  df-subrng 20462  df-subrg 20486  df-rlreg 20610  df-domn 20611  df-idom 20612  df-drng 20647  df-field 20648  df-lmod 20775  df-lss 20845  df-lsp 20885  df-cnfld 21272  df-zring 21364  df-zrh 21420  df-chr 21422  df-assa 21769  df-asp 21770  df-ascl 21771  df-psr 21825  df-mvr 21826  df-mpl 21827  df-opsr 21829  df-evls 21988  df-evl 21989  df-psr1 22071  df-vr1 22072  df-ply1 22073  df-coe1 22074  df-evl1 22210  df-mdeg 25967  df-deg1 25968  df-uc1p 26044  df-q1p 26045

This theorem is referenced by:  aks6d1c6lem3  42167