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Metamath Proof Explorer

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 ) ) ) )