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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cyc2fvx | Structured version Visualization version GIF version |
Description: Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
Ref | Expression |
---|---|
cycpm3.c | ⊢ 𝐶 = (toCyc‘𝐷) |
cycpm3.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
cycpm3.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
cycpm3.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
cycpm3.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
cycpm3.k | ⊢ (𝜑 → 𝐾 ∈ 𝐷) |
cycpm3.1 | ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
cycpm3.2 | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
cycpm3.3 | ⊢ (𝜑 → 𝐾 ≠ 𝐼) |
Ref | Expression |
---|---|
cyc2fvx | ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘𝐾) = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm3.c | . 2 ⊢ 𝐶 = (toCyc‘𝐷) | |
2 | cycpm3.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
3 | cycpm3.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | cycpm3.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
5 | 3, 4 | s2cld 14718 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽”〉 ∈ Word 𝐷) |
6 | cycpm3.1 | . . 3 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
7 | 3, 4, 6 | s2f1 31626 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽”〉:dom 〈“𝐼𝐽”〉–1-1→𝐷) |
8 | cycpm3.k | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
9 | cycpm3.3 | . . . 4 ⊢ (𝜑 → 𝐾 ≠ 𝐼) | |
10 | cycpm3.2 | . . . . 5 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
11 | 10 | necomd 2998 | . . . 4 ⊢ (𝜑 → 𝐾 ≠ 𝐽) |
12 | 9, 11 | nelprd 4616 | . . 3 ⊢ (𝜑 → ¬ 𝐾 ∈ {𝐼, 𝐽}) |
13 | 3, 4 | s2rn 31625 | . . 3 ⊢ (𝜑 → ran 〈“𝐼𝐽”〉 = {𝐼, 𝐽}) |
14 | 12, 13 | neleqtrrd 2861 | . 2 ⊢ (𝜑 → ¬ 𝐾 ∈ ran 〈“𝐼𝐽”〉) |
15 | 1, 2, 5, 7, 8, 14 | cycpmfv3 31789 | 1 ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘𝐾) = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 {cpr 4587 ran crn 5633 ‘cfv 6494 〈“cs2 14688 SymGrpcsymg 19107 toCycctocyc 31780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-sup 9337 df-inf 9338 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-n0 12373 df-z 12459 df-uz 12723 df-rp 12871 df-fz 13380 df-fzo 13523 df-fl 13652 df-mod 13730 df-hash 14185 df-word 14357 df-concat 14413 df-s1 14438 df-substr 14487 df-pfx 14517 df-csh 14635 df-s2 14695 df-tocyc 31781 |
This theorem is referenced by: cyc3co2 31814 |
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