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Theorem aks6d1c5 42121
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 (𝜑𝐾 ∈ Field)
aks6d1p5.2 (𝜑𝑃 ∈ ℙ)
aks6d1c5.3 𝑃 = (chr‘𝐾)
aks6d1c5.4 (𝜑𝐴 ∈ ℕ0)
aks6d1c5.5 (𝜑𝐴 < 𝑃)
aks6d1c5.6 𝑋 = (var1𝐾)
aks6d1c5.7 = (.g‘(mulGrp‘(Poly1𝐾)))
aks6d1c5.8 𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
Assertion
Ref Expression
aks6d1c5 (𝜑𝐺:(ℕ0m (0...𝐴))–1-1→(Base‘(Poly1𝐾)))
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.
StepHypRef Expression
1 eqid 2735 . . . . . 6 (Base‘(mulGrp‘(Poly1𝐾))) = (Base‘(mulGrp‘(Poly1𝐾)))
2 aks6d1p5.1 . . . . . . . . . 10 (𝜑𝐾 ∈ Field)
32fldcrngd 20759 . . . . . . . . 9 (𝜑𝐾 ∈ CRing)
4 eqid 2735 . . . . . . . . . 10 (Poly1𝐾) = (Poly1𝐾)
54ply1crng 22216 . . . . . . . . 9 (𝐾 ∈ CRing → (Poly1𝐾) ∈ CRing)
63, 5syl 17 . . . . . . . 8 (𝜑 → (Poly1𝐾) ∈ CRing)
7 eqid 2735 . . . . . . . . 9 (mulGrp‘(Poly1𝐾)) = (mulGrp‘(Poly1𝐾))
87crngmgp 20259 . . . . . . . 8 ((Poly1𝐾) ∈ CRing → (mulGrp‘(Poly1𝐾)) ∈ CMnd)
96, 8syl 17 . . . . . . 7 (𝜑 → (mulGrp‘(Poly1𝐾)) ∈ CMnd)
109adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (mulGrp‘(Poly1𝐾)) ∈ CMnd)
11 fzfid 14011 . . . . . 6 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (0...𝐴) ∈ Fin)
12 aks6d1c5.7 . . . . . . . 8 = (.g‘(mulGrp‘(Poly1𝐾)))
1310cmnmndd 19837 . . . . . . . . 9 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (mulGrp‘(Poly1𝐾)) ∈ Mnd)
1413adantr 480 . . . . . . . 8 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (mulGrp‘(Poly1𝐾)) ∈ Mnd)
15 nn0ex 12530 . . . . . . . . . . . . 13 0 ∈ V
1615a1i 11 . . . . . . . . . . . 12 (𝜑 → ℕ0 ∈ V)
17 ovexd 7466 . . . . . . . . . . . 12 (𝜑 → (0...𝐴) ∈ V)
1816, 17elmapd 8879 . . . . . . . . . . 11 (𝜑 → (𝑔 ∈ (ℕ0m (0...𝐴)) ↔ 𝑔:(0...𝐴)⟶ℕ0))
1918biimpd 229 . . . . . . . . . 10 (𝜑 → (𝑔 ∈ (ℕ0m (0...𝐴)) → 𝑔:(0...𝐴)⟶ℕ0))
2019imp 406 . . . . . . . . 9 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → 𝑔:(0...𝐴)⟶ℕ0)
2120ffvelcdmda 7104 . . . . . . . 8 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔𝑖) ∈ ℕ0)
226crngringd 20264 . . . . . . . . . . . . . 14 (𝜑 → (Poly1𝐾) ∈ Ring)
2322ringcmnd 20298 . . . . . . . . . . . . 13 (𝜑 → (Poly1𝐾) ∈ CMnd)
24 cmnmnd 19830 . . . . . . . . . . . . 13 ((Poly1𝐾) ∈ CMnd → (Poly1𝐾) ∈ Mnd)
2523, 24syl 17 . . . . . . . . . . . 12 (𝜑 → (Poly1𝐾) ∈ Mnd)
2625adantr 480 . . . . . . . . . . 11 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (Poly1𝐾) ∈ Mnd)
2726adantr 480 . . . . . . . . . 10 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (Poly1𝐾) ∈ Mnd)
283crngringd 20264 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Ring)
2928adantr 480 . . . . . . . . . . . 12 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → 𝐾 ∈ Ring)
3029adantr 480 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝐾 ∈ Ring)
31 aks6d1c5.6 . . . . . . . . . . . 12 𝑋 = (var1𝐾)
32 eqid 2735 . . . . . . . . . . . 12 (Base‘(Poly1𝐾)) = (Base‘(Poly1𝐾))
3331, 4, 32vr1cl 22235 . . . . . . . . . . 11 (𝐾 ∈ Ring → 𝑋 ∈ (Base‘(Poly1𝐾)))
3430, 33syl 17 . . . . . . . . . 10 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝑋 ∈ (Base‘(Poly1𝐾)))
35 simpl 482 . . . . . . . . . . . . 13 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝜑𝑔 ∈ (ℕ0m (0...𝐴))))
36 elfzelz 13561 . . . . . . . . . . . . . 14 (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ)
3736adantl 481 . . . . . . . . . . . . 13 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ ℤ)
3835, 37jca 511 . . . . . . . . . . . 12 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ ℤ))
39 eqid 2735 . . . . . . . . . . . . . . . 16 (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
4039zrhrhm 21540 . . . . . . . . . . . . . . 15 (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
41 zringbas 21482 . . . . . . . . . . . . . . . 16 ℤ = (Base‘ℤring)
42 eqid 2735 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
4341, 42rhmf 20502 . . . . . . . . . . . . . . 15 ((ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
4440, 43syl 17 . . . . . . . . . . . . . 14 (𝐾 ∈ Ring → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
4529, 44syl 17 . . . . . . . . . . . . 13 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
4645ffvelcdmda 7104 . . . . . . . . . . . 12 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ ℤ) → ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾))
4738, 46syl 17 . . . . . . . . . . 11 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾))
48 eqid 2735 . . . . . . . . . . . 12 (algSc‘(Poly1𝐾)) = (algSc‘(Poly1𝐾))
494, 48, 42, 32ply1sclcl 22305 . . . . . . . . . . 11 ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾)) → ((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1𝐾)))
5030, 47, 49syl2anc 584 . . . . . . . . . 10 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1𝐾)))
51 eqid 2735 . . . . . . . . . . 11 (+g‘(Poly1𝐾)) = (+g‘(Poly1𝐾))
5232, 51mndcl 18768 . . . . . . . . . 10 (((Poly1𝐾) ∈ Mnd ∧ 𝑋 ∈ (Base‘(Poly1𝐾)) ∧ ((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly1𝐾))) → (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly1𝐾)))
5327, 34, 50, 52syl3anc 1370 . . . . . . . . 9 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly1𝐾)))
547, 32mgpbas 20158 . . . . . . . . . 10 (Base‘(Poly1𝐾)) = (Base‘(mulGrp‘(Poly1𝐾)))
5554a1i 11 . . . . . . . . 9 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (Base‘(Poly1𝐾)) = (Base‘(mulGrp‘(Poly1𝐾))))
5653, 55eleqtrd 2841 . . . . . . . 8 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(mulGrp‘(Poly1𝐾))))
571, 12, 14, 21, 56mulgnn0cld 19126 . . . . . . 7 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(mulGrp‘(Poly1𝐾))))
5857ralrimiva 3144 . . . . . 6 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → ∀𝑖 ∈ (0...𝐴)((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(mulGrp‘(Poly1𝐾))))
591, 10, 11, 58gsummptcl 20000 . . . . 5 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(mulGrp‘(Poly1𝐾))))
6054eqcomi 2744 . . . . . 6 (Base‘(mulGrp‘(Poly1𝐾))) = (Base‘(Poly1𝐾))
6160a1i 11 . . . . 5 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (Base‘(mulGrp‘(Poly1𝐾))) = (Base‘(Poly1𝐾)))
6259, 61eleqtrd 2841 . . . 4 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(Poly1𝐾)))
63 aks6d1c5.8 . . . 4 𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
6462, 63fmptd 7134 . . 3 (𝜑𝐺:(ℕ0m (0...𝐴))⟶(Base‘(Poly1𝐾)))
65 eqidd 2736 . . . . . . . . 9 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (0g𝐾) = (0g𝐾))
66 simpr 484 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑥𝑦)
6766neneqd 2943 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ¬ 𝑥 = 𝑦)
68 simp-4r 784 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑥 ∈ (ℕ0m (0...𝐴)))
6915a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ℕ0 ∈ V)
70 ovexd 7466 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (0...𝐴) ∈ V)
7169, 70elmapd 8879 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (𝑥 ∈ (ℕ0m (0...𝐴)) ↔ 𝑥:(0...𝐴)⟶ℕ0))
7268, 71mpbid 232 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑥:(0...𝐴)⟶ℕ0)
73 ffn 6737 . . . . . . . . . . . . . . . . . . . 20 (𝑥:(0...𝐴)⟶ℕ0𝑥 Fn (0...𝐴))
7472, 73syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑥 Fn (0...𝐴))
75 simpllr 776 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑦 ∈ (ℕ0m (0...𝐴)))
7669, 70elmapd 8879 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (𝑦 ∈ (ℕ0m (0...𝐴)) ↔ 𝑦:(0...𝐴)⟶ℕ0))
7775, 76mpbid 232 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑦:(0...𝐴)⟶ℕ0)
78 ffn 6737 . . . . . . . . . . . . . . . . . . . 20 (𝑦:(0...𝐴)⟶ℕ0𝑦 Fn (0...𝐴))
7977, 78syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑦 Fn (0...𝐴))
80 eqfnfv2 7052 . . . . . . . . . . . . . . . . . . 19 ((𝑥 Fn (0...𝐴) ∧ 𝑦 Fn (0...𝐴)) → (𝑥 = 𝑦 ↔ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧))))
8174, 79, 80syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (𝑥 = 𝑦 ↔ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧))))
8281notbid 318 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (¬ 𝑥 = 𝑦 ↔ ¬ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧))))
8382biimpd 229 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (¬ 𝑥 = 𝑦 → ¬ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧))))
8467, 83mpd 15 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ¬ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧)))
85 ianor 983 . . . . . . . . . . . . . . 15 (¬ ((0...𝐴) = (0...𝐴) ∧ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧)) ↔ (¬ (0...𝐴) = (0...𝐴) ∨ ¬ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧)))
8684, 85sylib 218 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (¬ (0...𝐴) = (0...𝐴) ∨ ¬ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧)))
87 eqidd 2736 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (0...𝐴) = (0...𝐴))
8887notnotd 144 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ¬ ¬ (0...𝐴) = (0...𝐴))
8986, 88orcnd 878 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ¬ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧))
90 rexnal 3098 . . . . . . . . . . . . 13 (∃𝑧 ∈ (0...𝐴) ¬ (𝑥𝑧) = (𝑦𝑧) ↔ ¬ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧))
9189, 90sylibr 234 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ∃𝑧 ∈ (0...𝐴) ¬ (𝑥𝑧) = (𝑦𝑧))
92 df-ne 2939 . . . . . . . . . . . . 13 ((𝑥𝑧) ≠ (𝑦𝑧) ↔ ¬ (𝑥𝑧) = (𝑦𝑧))
9392rexbii 3092 . . . . . . . . . . . 12 (∃𝑧 ∈ (0...𝐴)(𝑥𝑧) ≠ (𝑦𝑧) ↔ ∃𝑧 ∈ (0...𝐴) ¬ (𝑥𝑧) = (𝑦𝑧))
9491, 93sylibr 234 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ∃𝑧 ∈ (0...𝐴)(𝑥𝑧) ≠ (𝑦𝑧))
95 simpl 482 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥𝑧) ≠ (𝑦𝑧))) → ((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦))
96 simprl 771 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥𝑧) ≠ (𝑦𝑧))) → 𝑧 ∈ (0...𝐴))
97 simprr 773 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥𝑧) ≠ (𝑦𝑧))) → (𝑥𝑧) ≠ (𝑦𝑧))
9895, 96, 97jca31 514 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥𝑧) ≠ (𝑦𝑧))) → ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) ≠ (𝑦𝑧)))
9971biimpd 229 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (𝑥 ∈ (ℕ0m (0...𝐴)) → 𝑥:(0...𝐴)⟶ℕ0))
10068, 99mpd 15 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑥:(0...𝐴)⟶ℕ0)
101100ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑥𝑧) ∈ ℕ0)
102101nn0red 12586 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑥𝑧) ∈ ℝ)
10376biimpd 229 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (𝑦 ∈ (ℕ0m (0...𝐴)) → 𝑦:(0...𝐴)⟶ℕ0))
10475, 103mpd 15 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → 𝑦:(0...𝐴)⟶ℕ0)
105104ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑦𝑧) ∈ ℕ0)
106105nn0red 12586 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑦𝑧) ∈ ℝ)
107102, 106lttri2d 11398 . . . . . . . . . . . . . 14 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) → ((𝑥𝑧) ≠ (𝑦𝑧) ↔ ((𝑥𝑧) < (𝑦𝑧) ∨ (𝑦𝑧) < (𝑥𝑧))))
1082ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → 𝐾 ∈ Field)
109 aks6d1p5.2 . . . . . . . . . . . . . . . . . 18 (𝜑𝑃 ∈ ℙ)
110109ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → 𝑃 ∈ ℙ)
111 aks6d1c5.3 . . . . . . . . . . . . . . . . 17 𝑃 = (chr‘𝐾)
112 aks6d1c5.4 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℕ0)
113112ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → 𝐴 ∈ ℕ0)
114 aks6d1c5.5 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 < 𝑃)
115114ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → 𝐴 < 𝑃)
11668ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → 𝑥 ∈ (ℕ0m (0...𝐴)))
11775ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → 𝑦 ∈ (ℕ0m (0...𝐴)))
118 simp-4r 784 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → (𝐺𝑥) = (𝐺𝑦))
119 simplr 769 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → 𝑧 ∈ (0...𝐴))
120 simpr 484 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → (𝑥𝑧) < (𝑦𝑧))
121108, 110, 111, 113, 115, 31, 12, 63, 116, 117, 118, 119, 120aks6d1c5lem2 42120 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) < (𝑦𝑧)) → (0g𝐾) ≠ (0g𝐾))
1222ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → 𝐾 ∈ Field)
123109ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → 𝑃 ∈ ℙ)
124112ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → 𝐴 ∈ ℕ0)
125114ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → 𝐴 < 𝑃)
12675ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → 𝑦 ∈ (ℕ0m (0...𝐴)))
12768ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → 𝑥 ∈ (ℕ0m (0...𝐴)))
128 simp-4r 784 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → (𝐺𝑥) = (𝐺𝑦))
129128eqcomd 2741 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → (𝐺𝑦) = (𝐺𝑥))
130 simplr 769 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → 𝑧 ∈ (0...𝐴))
131 simpr 484 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → (𝑦𝑧) < (𝑥𝑧))
132122, 123, 111, 124, 125, 31, 12, 63, 126, 127, 129, 130, 131aks6d1c5lem2 42120 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → (0g𝐾) ≠ (0g𝐾))
133121, 132jaodan 959 . . . . . . . . . . . . . . 15 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ ((𝑥𝑧) < (𝑦𝑧) ∨ (𝑦𝑧) < (𝑥𝑧))) → (0g𝐾) ≠ (0g𝐾))
134133ex 412 . . . . . . . . . . . . . 14 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (((𝑥𝑧) < (𝑦𝑧) ∨ (𝑦𝑧) < (𝑥𝑧)) → (0g𝐾) ≠ (0g𝐾)))
135107, 134sylbid 240 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) → ((𝑥𝑧) ≠ (𝑦𝑧) → (0g𝐾) ≠ (0g𝐾)))
136135imp 406 . . . . . . . . . . . 12 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑥𝑧) ≠ (𝑦𝑧)) → (0g𝐾) ≠ (0g𝐾))
13798, 136syl 17 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ (𝑧 ∈ (0...𝐴) ∧ (𝑥𝑧) ≠ (𝑦𝑧))) → (0g𝐾) ≠ (0g𝐾))
13894, 137rexlimddv 3159 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (0g𝐾) ≠ (0g𝐾))
139138neneqd 2943 . . . . . . . . 9 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ¬ (0g𝐾) = (0g𝐾))
14065, 139pm2.65da 817 . . . . . . . 8 ((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) → ¬ 𝑥𝑦)
141 df-ne 2939 . . . . . . . . 9 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
142141notbii 320 . . . . . . . 8 𝑥𝑦 ↔ ¬ ¬ 𝑥 = 𝑦)
143140, 142sylib 218 . . . . . . 7 ((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) → ¬ ¬ 𝑥 = 𝑦)
144 notnotb 315 . . . . . . 7 (𝑥 = 𝑦 ↔ ¬ ¬ 𝑥 = 𝑦)
145143, 144sylibr 234 . . . . . 6 ((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) → 𝑥 = 𝑦)
146145ex 412 . . . . 5 (((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) → ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦))
147146ralrimiva 3144 . . . 4 ((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) → ∀𝑦 ∈ (ℕ0m (0...𝐴))((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦))
148147ralrimiva 3144 . . 3 (𝜑 → ∀𝑥 ∈ (ℕ0m (0...𝐴))∀𝑦 ∈ (ℕ0m (0...𝐴))((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦))
14964, 148jca 511 . 2 (𝜑 → (𝐺:(ℕ0m (0...𝐴))⟶(Base‘(Poly1𝐾)) ∧ ∀𝑥 ∈ (ℕ0m (0...𝐴))∀𝑦 ∈ (ℕ0m (0...𝐴))((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)))
150 dff13 7275 . 2 (𝐺:(ℕ0m (0...𝐴))–1-1→(Base‘(Poly1𝐾)) ↔ (𝐺:(ℕ0m (0...𝐴))⟶(Base‘(Poly1𝐾)) ∧ ∀𝑥 ∈ (ℕ0m (0...𝐴))∀𝑦 ∈ (ℕ0m (0...𝐴))((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)))
151149, 150sylibr 234 1 (𝜑𝐺:(ℕ0m (0...𝐴))–1-1→(Base‘(Poly1𝐾)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478   class class class wbr 5148  cmpt 5231   Fn wfn 6558  wf 6559  1-1wf1 6560  cfv 6563  (class class class)co 7431  m cmap 8865  0cc0 11153   < clt 11293  0cn0 12524  cz 12611  ...cfz 13544  cprime 16705  Basecbs 17245  +gcplusg 17298  0gc0g 17486   Σg cgsu 17487  Mndcmnd 18760  .gcmg 19098  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252   RingHom crh 20486  Fieldcfield 20747  ringczring 21475  ℤRHomczrh 21528  chrcchr 21530  algSccascl 21890  var1cv1 22193  Poly1cpl1 22194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232  ax-mulf 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-prm 16706  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-od 19561  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-rhm 20489  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-rlreg 20711  df-domn 20712  df-idom 20713  df-drng 20748  df-field 20749  df-lmod 20877  df-lss 20948  df-lsp 20988  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-chr 21534  df-assa 21891  df-asp 21892  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-evls 22116  df-evl 22117  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-evl1 22336  df-mdeg 26109  df-deg1 26110  df-uc1p 26186  df-q1p 26187
This theorem is referenced by:  aks6d1c6lem3  42154
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