Theorem aks6d1c5 42112

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 2729	. . . . . 6 ⊢ (Base‘(mulGrp‘(Poly1‘𝐾))) = (Base‘(mulGrp‘(Poly1‘𝐾)))
2		aks6d1p5.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Field)
3	2	fldcrngd 20627	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CRing)
4		eqid 2729	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
5	4	ply1crng 22081	. . . . . . . . 9 ⊢ (𝐾 ∈ CRing → (Poly1‘𝐾) ∈ CRing)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CRing)
7		eqid 2729	. . . . . . . . 9 ⊢ (mulGrp‘(Poly1‘𝐾)) = (mulGrp‘(Poly1‘𝐾))
8	7	crngmgp 20126	. . . . . . . 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 13880	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) → (0...𝐴) ∈ Fin)
12		aks6d1c5.7	. . . . . . . 8 ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))
13	10	cmnmndd 19683	. . . . . . . . 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 12390	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ0 ∈ V)
17		ovexd 7384	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝐴) ∈ V)
18	16, 17	elmapd 8767	. . . . . . . . . . 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 7018	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℕ0 ↑m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) ∈ ℕ0)
22	6	crngringd 20131	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Ring)
23	22	ringcmnd 20169	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Poly1‘𝐾) ∈ CMnd)
24		cmnmnd 19676	. . . . . . . . . . . . 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 20131	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
33	31, 4, 32	vr1cl 22100	. . . . . . . . . . 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 13427	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
40	39	zrhrhm 21418	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
41		zringbas 21360	. . . . . . . . . . . . . . . 16 ⊢ ℤ = (Base‘ℤring)
42		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
43	41, 42	rhmf 20370	. . . . . . . . . . . . . . 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 7018	. . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝐾)) = (algSc‘(Poly1‘𝐾))
49	4, 48, 42, 32	ply1sclcl 22170	. . . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (+g‘(Poly1‘𝐾)) = (+g‘(Poly1‘𝐾))
52	32, 51	mndcl 18616	. . . . . . . . . 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 20030	. . . . . . . . . 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 2830	. . . . . . . 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 18974	. . . . . . 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 3121	. . . . . 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 19846	. . . . 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 2738	. . . . . 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 2830	. . . 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 7048	. . 3 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)))
65		eqidd 2730	. . . . . . . . 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 2930	. . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . . . . 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 8767	. . . . . . . . . . . . . . . . . . . . 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 6652	. . . . . . . . . . . . . . . . . . . 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 8767	. . . . . . . . . . . . . . . . . . . . 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 6652	. . . . . . . . . . . . . . . . . . . 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 6966	. . . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . 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 3081	. . . . . . . . . . . . 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 2926	. . . . . . . . . . . . 13 ⊢ ((𝑥‘𝑧) ≠ (𝑦‘𝑧) ↔ ¬ (𝑥‘𝑧) = (𝑦‘𝑧))
93	92	rexbii 3076	. . . . . . . . . . . 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 7018	. . . . . . . . . . . . . . . 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 12446	. . . . . . . . . . . . . . 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 7018	. . . . . . . . . . . . . . . 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 12446	. . . . . . . . . . . . . . 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 11255	. . . . . . . . . . . . . 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 42111	. . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . 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 42111	. . . . . . . . . . . . . . . 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 3136	. . . . . . . . . 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 2930	. . . . . . . . 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 2926	. . . . . . . . 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 3121	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℕ0 ↑m (0...𝐴))) → ∀𝑦 ∈ (ℕ0 ↑m (0...𝐴))((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
148	147	ralrimiva 3121	. . 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 7191	. 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 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3436   class class class wbr 5092   ↦ cmpt 5173   Fn wfn 6477  ⟶wf 6478  –1-1→wf1 6479  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  0cc0 11009   < clt 11149  ℕ₀cn0 12384  ℤcz 12471  ...cfz 13410  ℙcprime 16582  Basecbs 17120  +_gcplusg 17161  0_gc0g 17343   Σ_g cgsu 17344  Mndcmnd 18608  ._gcmg 18946  CMndccmn 19659  mulGrpcmgp 20025  Ringcrg 20118  CRingccrg 20119   RingHom crh 20354  Fieldcfield 20615  ℤ_ringczring 21353  ℤRHomczrh 21406  chrcchr 21408  algSccascl 21759  var₁cv1 22058  Poly₁cpl1 22059

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-addf 11088  ax-mulf 11089

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-ofr 7614  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-sup 9332  df-inf 9333  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-fl 13696  df-mod 13774  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-dvds 16164  df-prm 16583  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-ghm 19092  df-cntz 19196  df-od 19407  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-srg 20072  df-ring 20120  df-cring 20121  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-rhm 20357  df-nzr 20398  df-subrng 20431  df-subrg 20455  df-rlreg 20579  df-domn 20580  df-idom 20581  df-drng 20616  df-field 20617  df-lmod 20765  df-lss 20835  df-lsp 20875  df-cnfld 21262  df-zring 21354  df-zrh 21410  df-chr 21412  df-assa 21760  df-asp 21761  df-ascl 21762  df-psr 21816  df-mvr 21817  df-mpl 21818  df-opsr 21820  df-evls 21979  df-evl 21980  df-psr1 22062  df-vr1 22063  df-ply1 22064  df-coe1 22065  df-evl1 22201  df-mdeg 25958  df-deg1 25959  df-uc1p 26035  df-q1p 26036

This theorem is referenced by:  aks6d1c6lem3  42145