Users' Mathboxes Mathbox for metakunt < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aks6d1c5 Structured version   Visualization version   GIF version

Theorem aks6d1c5 42140
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 2737 . . . . . 6 (Base‘(mulGrp‘(Poly1𝐾))) = (Base‘(mulGrp‘(Poly1𝐾)))
2 aks6d1p5.1 . . . . . . . . . 10 (𝜑𝐾 ∈ Field)
32fldcrngd 20742 . . . . . . . . 9 (𝜑𝐾 ∈ CRing)
4 eqid 2737 . . . . . . . . . 10 (Poly1𝐾) = (Poly1𝐾)
54ply1crng 22200 . . . . . . . . 9 (𝐾 ∈ CRing → (Poly1𝐾) ∈ CRing)
63, 5syl 17 . . . . . . . 8 (𝜑 → (Poly1𝐾) ∈ CRing)
7 eqid 2737 . . . . . . . . 9 (mulGrp‘(Poly1𝐾)) = (mulGrp‘(Poly1𝐾))
87crngmgp 20238 . . . . . . . 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 14014 . . . . . 6 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (0...𝐴) ∈ Fin)
12 aks6d1c5.7 . . . . . . . 8 = (.g‘(mulGrp‘(Poly1𝐾)))
1310cmnmndd 19822 . . . . . . . . 9 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (mulGrp‘(Poly1𝐾)) ∈ Mnd)
1413adantr 480 . . . . . . . 8 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (mulGrp‘(Poly1𝐾)) ∈ Mnd)
15 nn0ex 12532 . . . . . . . . . . . . 13 0 ∈ V
1615a1i 11 . . . . . . . . . . . 12 (𝜑 → ℕ0 ∈ V)
17 ovexd 7466 . . . . . . . . . . . 12 (𝜑 → (0...𝐴) ∈ V)
1816, 17elmapd 8880 . . . . . . . . . . 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 20243 . . . . . . . . . . . . . 14 (𝜑 → (Poly1𝐾) ∈ Ring)
2322ringcmnd 20281 . . . . . . . . . . . . 13 (𝜑 → (Poly1𝐾) ∈ CMnd)
24 cmnmnd 19815 . . . . . . . . . . . . 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 20243 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . 12 (Base‘(Poly1𝐾)) = (Base‘(Poly1𝐾))
3331, 4, 32vr1cl 22219 . . . . . . . . . . 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 13564 . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . 16 (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
4039zrhrhm 21522 . . . . . . . . . . . . . . 15 (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
41 zringbas 21464 . . . . . . . . . . . . . . . 16 ℤ = (Base‘ℤring)
42 eqid 2737 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
4341, 42rhmf 20485 . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . 12 (algSc‘(Poly1𝐾)) = (algSc‘(Poly1𝐾))
494, 48, 42, 32ply1sclcl 22289 . . . . . . . . . . 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 2737 . . . . . . . . . . 11 (+g‘(Poly1𝐾)) = (+g‘(Poly1𝐾))
5232, 51mndcl 18755 . . . . . . . . . 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 1373 . . . . . . . . 9 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly1𝐾)))
547, 32mgpbas 20142 . . . . . . . . . 10 (Base‘(Poly1𝐾)) = (Base‘(mulGrp‘(Poly1𝐾)))
5554a1i 11 . . . . . . . . 9 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (Base‘(Poly1𝐾)) = (Base‘(mulGrp‘(Poly1𝐾))))
5653, 55eleqtrd 2843 . . . . . . . 8 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(mulGrp‘(Poly1𝐾))))
571, 12, 14, 21, 56mulgnn0cld 19113 . . . . . . 7 (((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(mulGrp‘(Poly1𝐾))))
5857ralrimiva 3146 . . . . . 6 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → ∀𝑖 ∈ (0...𝐴)((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(mulGrp‘(Poly1𝐾))))
591, 10, 11, 58gsummptcl 19985 . . . . 5 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖) (𝑋(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(mulGrp‘(Poly1𝐾))))
6054eqcomi 2746 . . . . . 6 (Base‘(mulGrp‘(Poly1𝐾))) = (Base‘(Poly1𝐾))
6160a1i 11 . . . . 5 ((𝜑𝑔 ∈ (ℕ0m (0...𝐴))) → (Base‘(mulGrp‘(Poly1𝐾))) = (Base‘(Poly1𝐾)))
6259, 61eleqtrd 2843 . . . 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 2738 . . . . . . . . 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 2945 . . . . . . . . . . . . . . . 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 8880 . . . . . . . . . . . . . . . . . . . . 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 6736 . . . . . . . . . . . . . . . . . . . 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 8880 . . . . . . . . . . . . . . . . . . . . 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 6736 . . . . . . . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 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 879 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → ¬ ∀𝑧 ∈ (0...𝐴)(𝑥𝑧) = (𝑦𝑧))
90 rexnal 3100 . . . . . . . . . . . . 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 2941 . . . . . . . . . . . . 13 ((𝑥𝑧) ≠ (𝑦𝑧) ↔ ¬ (𝑥𝑧) = (𝑦𝑧))
9392rexbii 3094 . . . . . . . . . . . 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 12588 . . . . . . . . . . . . . . 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 12588 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) → (𝑦𝑧) ∈ ℝ)
107102, 106lttri2d 11400 . . . . . . . . . . . . . 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 42139 . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . . 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 42139 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) ∧ 𝑧 ∈ (0...𝐴)) ∧ (𝑦𝑧) < (𝑥𝑧)) → (0g𝐾) ≠ (0g𝐾))
133121, 132jaodan 960 . . . . . . . . . . . . . . 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 3161 . . . . . . . . . 10 (((((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) ∧ 𝑦 ∈ (ℕ0m (0...𝐴))) ∧ (𝐺𝑥) = (𝐺𝑦)) ∧ 𝑥𝑦) → (0g𝐾) ≠ (0g𝐾))
139138neneqd 2945 . . . . . . . . 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 2941 . . . . . . . . 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 3146 . . . 4 ((𝜑𝑥 ∈ (ℕ0m (0...𝐴))) → ∀𝑦 ∈ (ℕ0m (0...𝐴))((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦))
148147ralrimiva 3146 . . 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 848   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480   class class class wbr 5143  cmpt 5225   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  m cmap 8866  0cc0 11155   < clt 11295  0cn0 12526  cz 12613  ...cfz 13547  cprime 16708  Basecbs 17247  +gcplusg 17297  0gc0g 17484   Σg cgsu 17485  Mndcmnd 18747  .gcmg 19085  CMndccmn 19798  mulGrpcmgp 20137  Ringcrg 20230  CRingccrg 20231   RingHom crh 20469  Fieldcfield 20730  ringczring 21457  ℤRHomczrh 21510  chrcchr 21512  algSccascl 21872  var1cv1 22177  Poly1cpl1 22178
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234  ax-mulf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-prm 16709  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-od 19546  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-rhm 20472  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-rlreg 20694  df-domn 20695  df-idom 20696  df-drng 20731  df-field 20732  df-lmod 20860  df-lss 20930  df-lsp 20970  df-cnfld 21365  df-zring 21458  df-zrh 21514  df-chr 21516  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-evl1 22320  df-mdeg 26094  df-deg1 26095  df-uc1p 26171  df-q1p 26172
This theorem is referenced by:  aks6d1c6lem3  42173
  Copyright terms: Public domain W3C validator