![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cyc3fv1 | Structured version Visualization version GIF version |
Description: Function value of a 3-cycle at the first point. (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 |
---|---|
cyc3fv1 | ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽𝐾”〉)‘𝐼) = 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm3.c | . . 3 ⊢ 𝐶 = (toCyc‘𝐷) | |
2 | cycpm3.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
3 | cycpm3.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | cycpm3.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
5 | cycpm3.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
6 | 3, 4, 5 | s3cld 14847 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ Word 𝐷) |
7 | cycpm3.1 | . . . 4 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
8 | cycpm3.2 | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
9 | cycpm3.3 | . . . 4 ⊢ (𝜑 → 𝐾 ≠ 𝐼) | |
10 | 3, 4, 5, 7, 8, 9 | s3f1 32652 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉:dom 〈“𝐼𝐽𝐾”〉–1-1→𝐷) |
11 | c0ex 11230 | . . . . . 6 ⊢ 0 ∈ V | |
12 | 11 | prid1 4762 | . . . . 5 ⊢ 0 ∈ {0, 1} |
13 | s3len 14869 | . . . . . . . . 9 ⊢ (♯‘〈“𝐼𝐽𝐾”〉) = 3 | |
14 | 13 | oveq1i 7424 | . . . . . . . 8 ⊢ ((♯‘〈“𝐼𝐽𝐾”〉) − 1) = (3 − 1) |
15 | 3m1e2 12362 | . . . . . . . 8 ⊢ (3 − 1) = 2 | |
16 | 14, 15 | eqtri 2755 | . . . . . . 7 ⊢ ((♯‘〈“𝐼𝐽𝐾”〉) − 1) = 2 |
17 | 16 | oveq2i 7425 | . . . . . 6 ⊢ (0..^((♯‘〈“𝐼𝐽𝐾”〉) − 1)) = (0..^2) |
18 | fzo0to2pr 13741 | . . . . . 6 ⊢ (0..^2) = {0, 1} | |
19 | 17, 18 | eqtri 2755 | . . . . 5 ⊢ (0..^((♯‘〈“𝐼𝐽𝐾”〉) − 1)) = {0, 1} |
20 | 12, 19 | eleqtrri 2827 | . . . 4 ⊢ 0 ∈ (0..^((♯‘〈“𝐼𝐽𝐾”〉) − 1)) |
21 | 20 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ (0..^((♯‘〈“𝐼𝐽𝐾”〉) − 1))) |
22 | 1, 2, 6, 10, 21 | cycpmfv1 32812 | . 2 ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽𝐾”〉)‘(〈“𝐼𝐽𝐾”〉‘0)) = (〈“𝐼𝐽𝐾”〉‘(0 + 1))) |
23 | s3fv0 14866 | . . . 4 ⊢ (𝐼 ∈ 𝐷 → (〈“𝐼𝐽𝐾”〉‘0) = 𝐼) | |
24 | 3, 23 | syl 17 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘0) = 𝐼) |
25 | 24 | fveq2d 6895 | . 2 ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽𝐾”〉)‘(〈“𝐼𝐽𝐾”〉‘0)) = ((𝐶‘〈“𝐼𝐽𝐾”〉)‘𝐼)) |
26 | 0p1e1 12356 | . . . 4 ⊢ (0 + 1) = 1 | |
27 | 26 | fveq2i 6894 | . . 3 ⊢ (〈“𝐼𝐽𝐾”〉‘(0 + 1)) = (〈“𝐼𝐽𝐾”〉‘1) |
28 | s3fv1 14867 | . . . 4 ⊢ (𝐽 ∈ 𝐷 → (〈“𝐼𝐽𝐾”〉‘1) = 𝐽) | |
29 | 4, 28 | syl 17 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘1) = 𝐽) |
30 | 27, 29 | eqtrid 2779 | . 2 ⊢ (𝜑 → (〈“𝐼𝐽𝐾”〉‘(0 + 1)) = 𝐽) |
31 | 22, 25, 30 | 3eqtr3d 2775 | 1 ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽𝐾”〉)‘𝐼) = 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 {cpr 4626 ‘cfv 6542 (class class class)co 7414 0cc0 11130 1c1 11131 + caddc 11133 − cmin 11466 2c2 12289 3c3 12290 ..^cfzo 13651 ♯chash 14313 〈“cs3 14817 SymGrpcsymg 19312 toCycctocyc 32805 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-hash 14314 df-word 14489 df-concat 14545 df-s1 14570 df-substr 14615 df-pfx 14645 df-csh 14763 df-s2 14823 df-s3 14824 df-tocyc 32806 |
This theorem is referenced by: cyc3co2 32839 |
Copyright terms: Public domain | W3C validator |