Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  psgnfzto1st Structured version   Visualization version   GIF version

Theorem psgnfzto1st 29632
Description: The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
Hypotheses
Ref Expression
psgnfzto1st.d 𝐷 = (1...𝑁)
psgnfzto1st.p 𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
psgnfzto1st.g 𝐺 = (SymGrp‘𝐷)
psgnfzto1st.b 𝐵 = (Base‘𝐺)
psgnfzto1st.s 𝑆 = (pmSgn‘𝐷)
Assertion
Ref Expression
psgnfzto1st (𝐼𝐷 → (𝑆𝑃) = (-1↑(𝐼 + 1)))
Distinct variable groups:   𝐷,𝑖   𝑖,𝐼   𝑖,𝑁   𝐵,𝑖
Allowed substitution hints:   𝑃(𝑖)   𝑆(𝑖)   𝐺(𝑖)

Proof of Theorem psgnfzto1st
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfz1b 12348 . . . . 5 (𝐼 ∈ (1...𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
21biimpi 206 . . . 4 (𝐼 ∈ (1...𝑁) → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
3 psgnfzto1st.d . . . 4 𝐷 = (1...𝑁)
42, 3eleq2s 2722 . . 3 (𝐼𝐷 → (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
5 3ancoma 1043 . . 3 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) ↔ (𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐼𝑁))
64, 5sylibr 224 . 2 (𝐼𝐷 → (𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁))
7 df-3an 1038 . . 3 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) ↔ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼𝑁))
8 breq1 4621 . . . . . 6 (𝑚 = 1 → (𝑚𝑁 ↔ 1 ≤ 𝑁))
9 id 22 . . . . . . . . . 10 (𝑚 = 1 → 𝑚 = 1)
10 breq2 4622 . . . . . . . . . . 11 (𝑚 = 1 → (𝑖𝑚𝑖 ≤ 1))
1110ifbid 4085 . . . . . . . . . 10 (𝑚 = 1 → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))
129, 11ifeq12d 4083 . . . . . . . . 9 (𝑚 = 1 → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
1312mpteq2dv 4710 . . . . . . . 8 (𝑚 = 1 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))))
1413fveq2d 6154 . . . . . . 7 (𝑚 = 1 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))))
15 oveq1 6612 . . . . . . . 8 (𝑚 = 1 → (𝑚 + 1) = (1 + 1))
1615oveq2d 6621 . . . . . . 7 (𝑚 = 1 → (-1↑(𝑚 + 1)) = (-1↑(1 + 1)))
1714, 16eqeq12d 2641 . . . . . 6 (𝑚 = 1 → ((𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1)) ↔ (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (-1↑(1 + 1))))
188, 17imbi12d 334 . . . . 5 (𝑚 = 1 → ((𝑚𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1))) ↔ (1 ≤ 𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (-1↑(1 + 1)))))
19 breq1 4621 . . . . . 6 (𝑚 = 𝑛 → (𝑚𝑁𝑛𝑁))
20 id 22 . . . . . . . . . 10 (𝑚 = 𝑛𝑚 = 𝑛)
21 breq2 4622 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑖𝑚𝑖𝑛))
2221ifbid 4085 . . . . . . . . . 10 (𝑚 = 𝑛 → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖𝑛, (𝑖 − 1), 𝑖))
2320, 22ifeq12d 4083 . . . . . . . . 9 (𝑚 = 𝑛 → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))
2423mpteq2dv 4710 . . . . . . . 8 (𝑚 = 𝑛 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))
2524fveq2d 6154 . . . . . . 7 (𝑚 = 𝑛 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))))
26 oveq1 6612 . . . . . . . 8 (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
2726oveq2d 6621 . . . . . . 7 (𝑚 = 𝑛 → (-1↑(𝑚 + 1)) = (-1↑(𝑛 + 1)))
2825, 27eqeq12d 2641 . . . . . 6 (𝑚 = 𝑛 → ((𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1)) ↔ (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1))))
2919, 28imbi12d 334 . . . . 5 (𝑚 = 𝑛 → ((𝑚𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1))) ↔ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))))
30 breq1 4621 . . . . . 6 (𝑚 = (𝑛 + 1) → (𝑚𝑁 ↔ (𝑛 + 1) ≤ 𝑁))
31 id 22 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
32 breq2 4622 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (𝑖𝑚𝑖 ≤ (𝑛 + 1)))
3332ifbid 4085 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))
3431, 33ifeq12d 4083 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))
3534mpteq2dv 4710 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))))
3635fveq2d 6154 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))))
37 oveq1 6612 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
3837oveq2d 6621 . . . . . . 7 (𝑚 = (𝑛 + 1) → (-1↑(𝑚 + 1)) = (-1↑((𝑛 + 1) + 1)))
3936, 38eqeq12d 2641 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1)) ↔ (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (-1↑((𝑛 + 1) + 1))))
4030, 39imbi12d 334 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝑚𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1))) ↔ ((𝑛 + 1) ≤ 𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (-1↑((𝑛 + 1) + 1)))))
41 breq1 4621 . . . . . 6 (𝑚 = 𝐼 → (𝑚𝑁𝐼𝑁))
42 id 22 . . . . . . . . . . 11 (𝑚 = 𝐼𝑚 = 𝐼)
43 breq2 4622 . . . . . . . . . . . 12 (𝑚 = 𝐼 → (𝑖𝑚𝑖𝐼))
4443ifbid 4085 . . . . . . . . . . 11 (𝑚 = 𝐼 → if(𝑖𝑚, (𝑖 − 1), 𝑖) = if(𝑖𝐼, (𝑖 − 1), 𝑖))
4542, 44ifeq12d 4083 . . . . . . . . . 10 (𝑚 = 𝐼 → if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)) = if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
4645mpteq2dv 4710 . . . . . . . . 9 (𝑚 = 𝐼 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖))))
47 psgnfzto1st.p . . . . . . . . 9 𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
4846, 47syl6eqr 2678 . . . . . . . 8 (𝑚 = 𝐼 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖))) = 𝑃)
4948fveq2d 6154 . . . . . . 7 (𝑚 = 𝐼 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (𝑆𝑃))
50 oveq1 6612 . . . . . . . 8 (𝑚 = 𝐼 → (𝑚 + 1) = (𝐼 + 1))
5150oveq2d 6621 . . . . . . 7 (𝑚 = 𝐼 → (-1↑(𝑚 + 1)) = (-1↑(𝐼 + 1)))
5249, 51eqeq12d 2641 . . . . . 6 (𝑚 = 𝐼 → ((𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1)) ↔ (𝑆𝑃) = (-1↑(𝐼 + 1))))
5341, 52imbi12d 334 . . . . 5 (𝑚 = 𝐼 → ((𝑚𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑚, if(𝑖𝑚, (𝑖 − 1), 𝑖)))) = (-1↑(𝑚 + 1))) ↔ (𝐼𝑁 → (𝑆𝑃) = (-1↑(𝐼 + 1)))))
54 fzfi 12708 . . . . . . . . 9 (1...𝑁) ∈ Fin
553, 54eqeltri 2700 . . . . . . . 8 𝐷 ∈ Fin
56 psgnfzto1st.s . . . . . . . . 9 𝑆 = (pmSgn‘𝐷)
5756psgnid 29624 . . . . . . . 8 (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)
5855, 57ax-mp 5 . . . . . . 7 (𝑆‘( I ↾ 𝐷)) = 1
59 eqid 2626 . . . . . . . . 9 1 = 1
60 eqid 2626 . . . . . . . . . 10 (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))
613, 60fzto1st1 29629 . . . . . . . . 9 (1 = 1 → (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷))
6259, 61ax-mp 5 . . . . . . . 8 (𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖))) = ( I ↾ 𝐷)
6362fveq2i 6153 . . . . . . 7 (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (𝑆‘( I ↾ 𝐷))
64 1p1e2 11079 . . . . . . . . 9 (1 + 1) = 2
6564oveq2i 6616 . . . . . . . 8 (-1↑(1 + 1)) = (-1↑2)
66 neg1sqe1 12896 . . . . . . . 8 (-1↑2) = 1
6765, 66eqtri 2648 . . . . . . 7 (-1↑(1 + 1)) = 1
6858, 63, 673eqtr4i 2658 . . . . . 6 (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (-1↑(1 + 1))
69682a1i 12 . . . . 5 (𝑁 ∈ ℕ → (1 ≤ 𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 1, if(𝑖 ≤ 1, (𝑖 − 1), 𝑖)))) = (-1↑(1 + 1))))
70 simplr 791 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℕ)
7170peano2nnd 10982 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℕ)
72 simpll 789 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
73 simpr 477 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ≤ 𝑁)
7471, 72, 733jca 1240 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
75 elfz1b 12348 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (1...𝑁) ↔ ((𝑛 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑛 + 1) ≤ 𝑁))
7674, 75sylibr 224 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ (1...𝑁))
7776, 3syl6eleqr 2715 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ 𝐷)
783psgnfzto1stlem 29627 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))))
7970, 77, 78syl2anc 692 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))))
8079adantlr 750 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))))
8180fveq2d 6154 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (𝑆‘(((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))))
8255a1i 11 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → 𝐷 ∈ Fin)
83 eqid 2626 . . . . . . . . . 10 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
84 psgnfzto1st.g . . . . . . . . . 10 𝐺 = (SymGrp‘𝐷)
85 psgnfzto1st.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
8683, 84, 85symgtrf 17805 . . . . . . . . 9 ran (pmTrsp‘𝐷) ⊆ 𝐵
87 eqid 2626 . . . . . . . . . . . 12 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
883, 87pmtrto1cl 29626 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ 𝐷) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
8970, 77, 88syl2anc 692 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
9089adantlr 750 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷))
9186, 90sseldi 3586 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵)
9270nnred 10980 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ ℝ)
93 1red 10000 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 1 ∈ ℝ)
9492, 93readdcld 10014 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 + 1) ∈ ℝ)
9572nnred 10980 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
9692lep1d 10900 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ≤ (𝑛 + 1))
9792, 94, 95, 96, 73letrd 10139 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝑁)
9870, 72, 973jca 1240 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑛𝑁))
99 elfz1b 12348 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑛𝑁))
10098, 99sylibr 224 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛 ∈ (1...𝑁))
101100, 3syl6eleqr 2715 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝐷)
102101adantlr 750 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝐷)
103 eqid 2626 . . . . . . . . . 10 (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) = (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))
1043, 103, 84, 85fzto1st 29630 . . . . . . . . 9 (𝑛𝐷 → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)
105102, 104syl 17 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵)
10684, 56, 85psgnco 19843 . . . . . . . 8 ((𝐷 ∈ Fin ∧ ((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ 𝐵 ∧ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))) ∈ 𝐵) → (𝑆‘(((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))) = ((𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) · (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))))
10782, 91, 105, 106syl3anc 1323 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘(((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))) = ((𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) · (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))))
10884, 83, 56psgnpmtr 17846 . . . . . . . . . . 11 (((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)}) ∈ ran (pmTrsp‘𝐷) → (𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) = -1)
10989, 108syl 17 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) = -1)
110109adantlr 750 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) = -1)
11197adantlr 750 . . . . . . . . . 10 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → 𝑛𝑁)
112 simplr 791 . . . . . . . . . 10 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1))))
113111, 112mpd 15 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))
114110, 113oveq12d 6623 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) · (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))) = (-1 · (-1↑(𝑛 + 1))))
115 neg1cn 11069 . . . . . . . . . . 11 -1 ∈ ℂ
116 peano2nn 10977 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
117116nnnn0d 11296 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ0)
118 expp1 12804 . . . . . . . . . . 11 ((-1 ∈ ℂ ∧ (𝑛 + 1) ∈ ℕ0) → (-1↑((𝑛 + 1) + 1)) = ((-1↑(𝑛 + 1)) · -1))
119115, 117, 118sylancr 694 . . . . . . . . . 10 (𝑛 ∈ ℕ → (-1↑((𝑛 + 1) + 1)) = ((-1↑(𝑛 + 1)) · -1))
120115a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → -1 ∈ ℂ)
121120, 117expcld 12945 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (-1↑(𝑛 + 1)) ∈ ℂ)
122121, 120mulcomd 10006 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((-1↑(𝑛 + 1)) · -1) = (-1 · (-1↑(𝑛 + 1))))
123119, 122eqtr2d 2661 . . . . . . . . 9 (𝑛 ∈ ℕ → (-1 · (-1↑(𝑛 + 1))) = (-1↑((𝑛 + 1) + 1)))
124123ad3antlr 766 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (-1 · (-1↑(𝑛 + 1))) = (-1↑((𝑛 + 1) + 1)))
125114, 124eqtrd 2660 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → ((𝑆‘((pmTrsp‘𝐷)‘{𝑛, (𝑛 + 1)})) · (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖))))) = (-1↑((𝑛 + 1) + 1)))
12681, 107, 1253eqtrd 2664 . . . . . 6 ((((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) ∧ (𝑛 + 1) ≤ 𝑁) → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (-1↑((𝑛 + 1) + 1)))
127126ex 450 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑛𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, 𝑛, if(𝑖𝑛, (𝑖 − 1), 𝑖)))) = (-1↑(𝑛 + 1)))) → ((𝑛 + 1) ≤ 𝑁 → (𝑆‘(𝑖𝐷 ↦ if(𝑖 = 1, (𝑛 + 1), if(𝑖 ≤ (𝑛 + 1), (𝑖 − 1), 𝑖)))) = (-1↑((𝑛 + 1) + 1))))
12818, 29, 40, 53, 69, 127nnindd 29399 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) → (𝐼𝑁 → (𝑆𝑃) = (-1↑(𝐼 + 1))))
129128imp 445 . . 3 (((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ) ∧ 𝐼𝑁) → (𝑆𝑃) = (-1↑(𝐼 + 1)))
1307, 129sylbi 207 . 2 ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼𝑁) → (𝑆𝑃) = (-1↑(𝐼 + 1)))
1316, 130syl 17 1 (𝐼𝐷 → (𝑆𝑃) = (-1↑(𝐼 + 1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  ifcif 4063  {cpr 4155   class class class wbr 4618  cmpt 4678   I cid 4989  ran crn 5080  cres 5081  ccom 5083  cfv 5850  (class class class)co 6605  Fincfn 7900  cc 9879  1c1 9882   + caddc 9884   · cmul 9886  cle 10020  cmin 10211  -cneg 10212  cn 10965  2c2 11015  0cn0 11237  ...cfz 12265  cexp 12797  Basecbs 15776  SymGrpcsymg 17713  pmTrspcpmtr 17777  pmSgncpsgn 17825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-addf 9960  ax-mulf 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-xnn0 11309  df-z 11323  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-word 13233  df-lsw 13234  df-concat 13235  df-s1 13236  df-substr 13237  df-splice 13238  df-reverse 13239  df-s2 13525  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-starv 15872  df-tset 15876  df-ple 15877  df-ds 15880  df-unif 15881  df-0g 16018  df-gsum 16019  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-subg 17507  df-ghm 17574  df-gim 17617  df-oppg 17692  df-symg 17714  df-pmtr 17778  df-psgn 17827  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-cring 18466  df-oppr 18539  df-dvdsr 18557  df-unit 18558  df-invr 18588  df-dvr 18599  df-drng 18665  df-cnfld 19661
This theorem is referenced by:  madjusmdetlem4  29670
  Copyright terms: Public domain W3C validator