Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cyc2fv2 | Structured version Visualization version GIF version |
Description: Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
Ref | Expression |
---|---|
cycpm2.c | ⊢ 𝐶 = (toCyc‘𝐷) |
cycpm2.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
cycpm2.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
cycpm2.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
cycpm2.1 | ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
cycpm2cl.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
Ref | Expression |
---|---|
cyc2fv2 | ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘𝐽) = 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm2.c | . . 3 ⊢ 𝐶 = (toCyc‘𝐷) | |
2 | cycpm2.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
3 | cycpm2.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | cycpm2.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
5 | 3, 4 | s2cld 14226 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽”〉 ∈ Word 𝐷) |
6 | cycpm2.1 | . . . 4 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
7 | 3, 4, 6 | s2f1 30619 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽”〉:dom 〈“𝐼𝐽”〉–1-1→𝐷) |
8 | 2pos 11734 | . . . . 5 ⊢ 0 < 2 | |
9 | s2len 14244 | . . . . 5 ⊢ (♯‘〈“𝐼𝐽”〉) = 2 | |
10 | 8, 9 | breqtrri 5086 | . . . 4 ⊢ 0 < (♯‘〈“𝐼𝐽”〉) |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → 0 < (♯‘〈“𝐼𝐽”〉)) |
12 | 9 | oveq1i 7159 | . . . . 5 ⊢ ((♯‘〈“𝐼𝐽”〉) − 1) = (2 − 1) |
13 | 2m1e1 11757 | . . . . 5 ⊢ (2 − 1) = 1 | |
14 | 12, 13 | eqtr2i 2844 | . . . 4 ⊢ 1 = ((♯‘〈“𝐼𝐽”〉) − 1) |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → 1 = ((♯‘〈“𝐼𝐽”〉) − 1)) |
16 | 1, 2, 5, 7, 11, 15 | cycpmfv2 30775 | . 2 ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘(〈“𝐼𝐽”〉‘1)) = (〈“𝐼𝐽”〉‘0)) |
17 | s2fv1 14243 | . . . 4 ⊢ (𝐽 ∈ 𝐷 → (〈“𝐼𝐽”〉‘1) = 𝐽) | |
18 | 4, 17 | syl 17 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽”〉‘1) = 𝐽) |
19 | 18 | fveq2d 6667 | . 2 ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘(〈“𝐼𝐽”〉‘1)) = ((𝐶‘〈“𝐼𝐽”〉)‘𝐽)) |
20 | s2fv0 14242 | . . 3 ⊢ (𝐼 ∈ 𝐷 → (〈“𝐼𝐽”〉‘0) = 𝐼) | |
21 | 3, 20 | syl 17 | . 2 ⊢ (𝜑 → (〈“𝐼𝐽”〉‘0) = 𝐼) |
22 | 16, 19, 21 | 3eqtr3d 2863 | 1 ⊢ (𝜑 → ((𝐶‘〈“𝐼𝐽”〉)‘𝐽) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 0cc0 10530 1c1 10531 < clt 10668 − cmin 10863 2c2 11686 ♯chash 13687 〈“cs2 14196 SymGrpcsymg 18488 toCycctocyc 30767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-fl 13159 df-mod 13235 df-hash 13688 df-word 13859 df-concat 13916 df-s1 13943 df-substr 13996 df-pfx 14026 df-csh 14144 df-s2 14203 df-tocyc 30768 |
This theorem is referenced by: cycpmco2 30794 cyc3co2 30801 |
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