Theorem pgphash

Description: The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016)

	Ref	Expression
Hypothesis	pgpfi.1	⊢ 𝑋 = ( Base ‘ 𝐺 )
Assertion	pgphash	⊢ ( ( 𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin ) → ( ♯ ‘ 𝑋 ) = ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		pgpfi.1	⊢ 𝑋 = ( Base ‘ 𝐺 )
2		simpl	⊢ ( ( 𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin ) → 𝑃 pGrp 𝐺 )
3		pgpgrp	⊢ ( 𝑃 pGrp 𝐺 → 𝐺 ∈ Grp )
4	1	pgpfi2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ) → ( 𝑃 pGrp 𝐺 ↔ ( 𝑃 ∈ ℙ ∧ ( ♯ ‘ 𝑋 ) = ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) ) ) )
5	3 4	sylan	⊢ ( ( 𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin ) → ( 𝑃 pGrp 𝐺 ↔ ( 𝑃 ∈ ℙ ∧ ( ♯ ‘ 𝑋 ) = ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) ) ) )
6	2 5	mpbid	⊢ ( ( 𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin ) → ( 𝑃 ∈ ℙ ∧ ( ♯ ‘ 𝑋 ) = ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) ) )
7	6	simprd	⊢ ( ( 𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin ) → ( ♯ ‘ 𝑋 ) = ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) )