Theorem fzto1stinvn

Description: Value of the inverse of our permutation P at I . (Contributed by Thierry Arnoux, 23-Aug-2020)

	Ref	Expression
Hypotheses	psgnfzto1st.d	⊢ 𝐷 = ( 1 ... 𝑁 )
	psgnfzto1st.p	⊢ 𝑃 = ( 𝑖 ∈ 𝐷 ↦ if ( 𝑖 = 1 , 𝐼 , if ( 𝑖 ≤ 𝐼 , ( 𝑖 − 1 ) , 𝑖 ) ) )
	psgnfzto1st.g	⊢ 𝐺 = ( SymGrp ‘ 𝐷 )
	psgnfzto1st.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	fzto1stinvn	⊢ ( 𝐼 ∈ 𝐷 → ( ◡ 𝑃 ‘ 𝐼 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		psgnfzto1st.d	⊢ 𝐷 = ( 1 ... 𝑁 )
2		psgnfzto1st.p	⊢ 𝑃 = ( 𝑖 ∈ 𝐷 ↦ if ( 𝑖 = 1 , 𝐼 , if ( 𝑖 ≤ 𝐼 , ( 𝑖 − 1 ) , 𝑖 ) ) )
3		psgnfzto1st.g	⊢ 𝐺 = ( SymGrp ‘ 𝐷 )
4		psgnfzto1st.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
5	1 2	fzto1stfv1	⊢ ( 𝐼 ∈ 𝐷 → ( 𝑃 ‘ 1 ) = 𝐼 )
6	5	fveq2d	⊢ ( 𝐼 ∈ 𝐷 → ( ◡ 𝑃 ‘ ( 𝑃 ‘ 1 ) ) = ( ◡ 𝑃 ‘ 𝐼 ) )
7	1 2 3 4	fzto1st	⊢ ( 𝐼 ∈ 𝐷 → 𝑃 ∈ 𝐵 )
8	3 4	symgbasf1o	⊢ ( 𝑃 ∈ 𝐵 → 𝑃 : 𝐷 –1-1-onto→ 𝐷 )
9	7 8	syl	⊢ ( 𝐼 ∈ 𝐷 → 𝑃 : 𝐷 –1-1-onto→ 𝐷 )
10		elfzuz2	⊢ ( 𝐼 ∈ ( 1 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 1 ) )
11	10 1	eleq2s	⊢ ( 𝐼 ∈ 𝐷 → 𝑁 ∈ ( ℤ≥ ‘ 1 ) )
12		eluzfz1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑁 ) )
13	12 1	eleqtrrdi	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ 𝐷 )
14	11 13	syl	⊢ ( 𝐼 ∈ 𝐷 → 1 ∈ 𝐷 )
15		f1ocnvfv1	⊢ ( ( 𝑃 : 𝐷 –1-1-onto→ 𝐷 ∧ 1 ∈ 𝐷 ) → ( ◡ 𝑃 ‘ ( 𝑃 ‘ 1 ) ) = 1 )
16	9 14 15	syl2anc	⊢ ( 𝐼 ∈ 𝐷 → ( ◡ 𝑃 ‘ ( 𝑃 ‘ 1 ) ) = 1 )
17	6 16	eqtr3d	⊢ ( 𝐼 ∈ 𝐷 → ( ◡ 𝑃 ‘ 𝐼 ) = 1 )