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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fzto1stinvn | Structured version Visualization version GIF version |
Description: Value of the inverse of our permutation 𝑃 at 𝐼. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
Ref | Expression |
---|---|
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) |
Ref | Expression |
---|---|
fzto1stinvn | ⊢ (𝐼 ∈ 𝐷 → (◡𝑃‘𝐼) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgnfzto1st.d | . . . 4 ⊢ 𝐷 = (1...𝑁) | |
2 | psgnfzto1st.p | . . . 4 ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) | |
3 | 1, 2 | fzto1stfv1 32835 | . . 3 ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) |
4 | 3 | fveq2d 6901 | . 2 ⊢ (𝐼 ∈ 𝐷 → (◡𝑃‘(𝑃‘1)) = (◡𝑃‘𝐼)) |
5 | psgnfzto1st.g | . . . . 5 ⊢ 𝐺 = (SymGrp‘𝐷) | |
6 | psgnfzto1st.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
7 | 1, 2, 5, 6 | fzto1st 32837 | . . . 4 ⊢ (𝐼 ∈ 𝐷 → 𝑃 ∈ 𝐵) |
8 | 5, 6 | symgbasf1o 19329 | . . . 4 ⊢ (𝑃 ∈ 𝐵 → 𝑃:𝐷–1-1-onto→𝐷) |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐼 ∈ 𝐷 → 𝑃:𝐷–1-1-onto→𝐷) |
10 | elfzuz2 13539 | . . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1)) | |
11 | 10, 1 | eleq2s 2847 | . . . 4 ⊢ (𝐼 ∈ 𝐷 → 𝑁 ∈ (ℤ≥‘1)) |
12 | eluzfz1 13541 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
13 | 12, 1 | eleqtrrdi 2840 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ 𝐷) |
14 | 11, 13 | syl 17 | . . 3 ⊢ (𝐼 ∈ 𝐷 → 1 ∈ 𝐷) |
15 | f1ocnvfv1 7285 | . . 3 ⊢ ((𝑃:𝐷–1-1-onto→𝐷 ∧ 1 ∈ 𝐷) → (◡𝑃‘(𝑃‘1)) = 1) | |
16 | 9, 14, 15 | syl2anc 583 | . 2 ⊢ (𝐼 ∈ 𝐷 → (◡𝑃‘(𝑃‘1)) = 1) |
17 | 4, 16 | eqtr3d 2770 | 1 ⊢ (𝐼 ∈ 𝐷 → (◡𝑃‘𝐼) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ifcif 4529 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5677 –1-1-onto→wf1o 6547 ‘cfv 6548 (class class class)co 7420 1c1 11140 ≤ cle 11280 − cmin 11475 ℤ≥cuz 12853 ...cfz 13517 Basecbs 17180 SymGrpcsymg 19321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-tset 17252 df-efmnd 18821 df-symg 19322 df-pmtr 19397 |
This theorem is referenced by: madjusmdetlem4 33431 |
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