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Theorem isprimroot 41998
Description: The value of a primitive root. (Contributed by metakunt, 25-Apr-2025.)
Hypotheses
Ref Expression
isprimroot.1 (𝜑𝑅 ∈ CMnd)
isprimroot.2 (𝜑𝐾 ∈ ℕ0)
isprimroot.3 = (.g𝑅)
Assertion
Ref Expression
isprimroot (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 𝑀) = (0g𝑅) → 𝐾𝑙))))
Distinct variable groups:   𝐾,𝑙   𝑀,𝑙   𝑅,𝑙   𝜑,𝑙
Allowed substitution hint:   (𝑙)

Proof of Theorem isprimroot
Dummy variables 𝑏 𝑘 𝑟 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-primroots 41997 . . . . . 6 PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ0(Base‘𝑟) / 𝑏{𝑥𝑏 ∣ ((𝑘(.g𝑟)𝑥) = (0g𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑟)𝑥) = (0g𝑟) → 𝑘𝑙))})
21a1i 11 . . . . 5 (𝜑 → PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ0(Base‘𝑟) / 𝑏{𝑥𝑏 ∣ ((𝑘(.g𝑟)𝑥) = (0g𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑟)𝑥) = (0g𝑟) → 𝑘𝑙))}))
3 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) → 𝑟 = 𝑅)
43fveq2d 6923 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) → (Base‘𝑟) = (Base‘𝑅))
5 simplrl 776 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → 𝑟 = 𝑅)
65fveq2d 6923 . . . . . . . . . 10 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → (.g𝑟) = (.g𝑅))
7 simplrr 777 . . . . . . . . . 10 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → 𝑘 = 𝐾)
8 eqidd 2735 . . . . . . . . . 10 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → 𝑥 = 𝑥)
96, 7, 8oveq123d 7466 . . . . . . . . 9 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → (𝑘(.g𝑟)𝑥) = (𝐾(.g𝑅)𝑥))
105fveq2d 6923 . . . . . . . . 9 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → (0g𝑟) = (0g𝑅))
119, 10eqeq12d 2750 . . . . . . . 8 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → ((𝑘(.g𝑟)𝑥) = (0g𝑟) ↔ (𝐾(.g𝑅)𝑥) = (0g𝑅)))
123fveq2d 6923 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) → (.g𝑟) = (.g𝑅))
1312oveqdr 7473 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → (𝑙(.g𝑟)𝑥) = (𝑙(.g𝑅)𝑥))
1413, 10eqeq12d 2750 . . . . . . . . . 10 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → ((𝑙(.g𝑟)𝑥) = (0g𝑟) ↔ (𝑙(.g𝑅)𝑥) = (0g𝑅)))
157breq1d 5179 . . . . . . . . . 10 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → (𝑘𝑙𝐾𝑙))
1614, 15imbi12d 344 . . . . . . . . 9 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → (((𝑙(.g𝑟)𝑥) = (0g𝑟) → 𝑘𝑙) ↔ ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙)))
1716ralbidv 3180 . . . . . . . 8 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → (∀𝑙 ∈ ℕ0 ((𝑙(.g𝑟)𝑥) = (0g𝑟) → 𝑘𝑙) ↔ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙)))
1811, 17anbi12d 631 . . . . . . 7 (((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) ∧ 𝑥𝑏) → (((𝑘(.g𝑟)𝑥) = (0g𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑟)𝑥) = (0g𝑟) → 𝑘𝑙)) ↔ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))))
1918rabbidva 3445 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) → {𝑥𝑏 ∣ ((𝑘(.g𝑟)𝑥) = (0g𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑟)𝑥) = (0g𝑟) → 𝑘𝑙))} = {𝑥𝑏 ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))})
204, 19csbeq12dv 3924 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑘 = 𝐾)) → (Base‘𝑟) / 𝑏{𝑥𝑏 ∣ ((𝑘(.g𝑟)𝑥) = (0g𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑟)𝑥) = (0g𝑟) → 𝑘𝑙))} = (Base‘𝑅) / 𝑏{𝑥𝑏 ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))})
21 isprimroot.1 . . . . 5 (𝜑𝑅 ∈ CMnd)
22 isprimroot.2 . . . . 5 (𝜑𝐾 ∈ ℕ0)
23 eqid 2734 . . . . . . 7 {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} = {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))}
24 fvexd 6934 . . . . . . 7 (𝜑 → (Base‘𝑅) ∈ V)
2523, 24rabexd 5361 . . . . . 6 (𝜑 → {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} ∈ V)
26 simpr 484 . . . . . . . . 9 ((𝜑𝑏 = (Base‘𝑅)) → 𝑏 = (Base‘𝑅))
2726rabeqdv 3454 . . . . . . . 8 ((𝜑𝑏 = (Base‘𝑅)) → {𝑥𝑏 ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} = {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))})
2824, 27csbied 3953 . . . . . . 7 (𝜑(Base‘𝑅) / 𝑏{𝑥𝑏 ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} = {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))})
2928eleq1d 2823 . . . . . 6 (𝜑 → ((Base‘𝑅) / 𝑏{𝑥𝑏 ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} ∈ V ↔ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} ∈ V))
3025, 29mpbird 257 . . . . 5 (𝜑(Base‘𝑅) / 𝑏{𝑥𝑏 ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} ∈ V)
312, 20, 21, 22, 30ovmpod 7598 . . . 4 (𝜑 → (𝑅 PrimRoots 𝐾) = (Base‘𝑅) / 𝑏{𝑥𝑏 ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))})
3231, 28eqtrd 2774 . . 3 (𝜑 → (𝑅 PrimRoots 𝐾) = {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))})
3332eleq2d 2824 . 2 (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ 𝑀 ∈ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))}))
34 oveq2 7453 . . . . . . 7 (𝑥 = 𝑀 → (𝐾(.g𝑅)𝑥) = (𝐾(.g𝑅)𝑀))
3534eqeq1d 2736 . . . . . 6 (𝑥 = 𝑀 → ((𝐾(.g𝑅)𝑥) = (0g𝑅) ↔ (𝐾(.g𝑅)𝑀) = (0g𝑅)))
36 oveq2 7453 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑙(.g𝑅)𝑥) = (𝑙(.g𝑅)𝑀))
3736eqeq1d 2736 . . . . . . . 8 (𝑥 = 𝑀 → ((𝑙(.g𝑅)𝑥) = (0g𝑅) ↔ (𝑙(.g𝑅)𝑀) = (0g𝑅)))
3837imbi1d 341 . . . . . . 7 (𝑥 = 𝑀 → (((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙) ↔ ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙)))
3938ralbidv 3180 . . . . . 6 (𝑥 = 𝑀 → (∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙) ↔ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙)))
4035, 39anbi12d 631 . . . . 5 (𝑥 = 𝑀 → (((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙)) ↔ ((𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙))))
4140elrab 3703 . . . 4 (𝑀 ∈ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} ↔ (𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙))))
4241a1i 11 . . 3 (𝜑 → (𝑀 ∈ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} ↔ (𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙)))))
43 3anass 1095 . . . . . 6 ((𝑀 ∈ (Base‘𝑅) ∧ (𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙)) ↔ (𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙))))
4443bicomi 224 . . . . 5 ((𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙))) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙)))
4544a1i 11 . . . 4 (𝜑 → ((𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙))) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙))))
46 biidd 262 . . . . 5 (𝜑 → (𝑀 ∈ (Base‘𝑅) ↔ 𝑀 ∈ (Base‘𝑅)))
47 isprimroot.3 . . . . . . . . 9 = (.g𝑅)
4847eqcomi 2743 . . . . . . . 8 (.g𝑅) =
4948a1i 11 . . . . . . 7 (𝜑 → (.g𝑅) = )
5049oveqd 7462 . . . . . 6 (𝜑 → (𝐾(.g𝑅)𝑀) = (𝐾 𝑀))
5150eqeq1d 2736 . . . . 5 (𝜑 → ((𝐾(.g𝑅)𝑀) = (0g𝑅) ↔ (𝐾 𝑀) = (0g𝑅)))
5249oveqd 7462 . . . . . . . 8 (𝜑 → (𝑙(.g𝑅)𝑀) = (𝑙 𝑀))
5352eqeq1d 2736 . . . . . . 7 (𝜑 → ((𝑙(.g𝑅)𝑀) = (0g𝑅) ↔ (𝑙 𝑀) = (0g𝑅)))
5453imbi1d 341 . . . . . 6 (𝜑 → (((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙) ↔ ((𝑙 𝑀) = (0g𝑅) → 𝐾𝑙)))
5554ralbidv 3180 . . . . 5 (𝜑 → (∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙) ↔ ∀𝑙 ∈ ℕ0 ((𝑙 𝑀) = (0g𝑅) → 𝐾𝑙)))
5646, 51, 553anbi123d 1436 . . . 4 (𝜑 → ((𝑀 ∈ (Base‘𝑅) ∧ (𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙)) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 𝑀) = (0g𝑅) → 𝐾𝑙))))
5745, 56bitrd 279 . . 3 (𝜑 → ((𝑀 ∈ (Base‘𝑅) ∧ ((𝐾(.g𝑅)𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑀) = (0g𝑅) → 𝐾𝑙))) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 𝑀) = (0g𝑅) → 𝐾𝑙))))
5842, 57bitrd 279 . 2 (𝜑 → (𝑀 ∈ {𝑥 ∈ (Base‘𝑅) ∣ ((𝐾(.g𝑅)𝑥) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g𝑅)𝑥) = (0g𝑅) → 𝐾𝑙))} ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 𝑀) = (0g𝑅) → 𝐾𝑙))))
5933, 58bitrd 279 1 (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 𝑀) = (0g𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 𝑀) = (0g𝑅) → 𝐾𝑙))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wral 3063  {crab 3438  Vcvv 3482  csb 3915   class class class wbr 5169  cfv 6572  (class class class)co 7445  cmpo 7447  0cn0 12549  cdvds 16296  Basecbs 17253  0gc0g 17494  .gcmg 19102  CMndccmn 19817   PrimRoots cprimroots 41996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-primroots 41997
This theorem is referenced by:  isprimroot2  41999  primrootsunit1  42002  primrootscoprmpow  42004  primrootscoprbij  42007  primrootlekpowne0  42010  primrootspoweq0  42011  aks6d1c1p2  42014  aks6d1c1p3  42015  aks6d1c1p4  42016  aks6d1c1p5  42017  aks6d1c1p7  42018  aks6d1c1p6  42019  aks6d1c1p8  42020  aks6d1c2lem3  42031  aks6d1c2lem4  42032  aks6d1c6lem2  42076  aks6d1c6lem3  42077  aks6d1c6lem4  42078  aks6d1c6isolem1  42079  aks6d1c6isolem2  42080  aks6d1c6lem5  42082  aks5lem2  42092  aks5lem3a  42094
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