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Mirrors > Home > MPE Home > Th. List > revs1 | Structured version Visualization version GIF version |
Description: Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
revs1 | ⊢ (reverse‘〈“𝑆”〉) = 〈“𝑆”〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cli 14127 | . . . . 5 ⊢ 〈“𝑆”〉 ∈ Word V | |
2 | s1len 14128 | . . . . . . 7 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 1nn 11806 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2, 3 | eqeltri 2827 | . . . . . 6 ⊢ (♯‘〈“𝑆”〉) ∈ ℕ |
5 | lbfzo0 13247 | . . . . . 6 ⊢ (0 ∈ (0..^(♯‘〈“𝑆”〉)) ↔ (♯‘〈“𝑆”〉) ∈ ℕ) | |
6 | 4, 5 | mpbir 234 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘〈“𝑆”〉)) |
7 | revfv 14293 | . . . . 5 ⊢ ((〈“𝑆”〉 ∈ Word V ∧ 0 ∈ (0..^(♯‘〈“𝑆”〉))) → ((reverse‘〈“𝑆”〉)‘0) = (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0))) | |
8 | 1, 6, 7 | mp2an 692 | . . . 4 ⊢ ((reverse‘〈“𝑆”〉)‘0) = (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) |
9 | 2 | oveq1i 7201 | . . . . . . . . 9 ⊢ ((♯‘〈“𝑆”〉) − 1) = (1 − 1) |
10 | 1m1e0 11867 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
11 | 9, 10 | eqtri 2759 | . . . . . . . 8 ⊢ ((♯‘〈“𝑆”〉) − 1) = 0 |
12 | 11 | oveq1i 7201 | . . . . . . 7 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = (0 − 0) |
13 | 0m0e0 11915 | . . . . . . 7 ⊢ (0 − 0) = 0 | |
14 | 12, 13 | eqtri 2759 | . . . . . 6 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = 0 |
15 | 14 | fveq2i 6698 | . . . . 5 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = (〈“𝑆”〉‘0) |
16 | ids1 14119 | . . . . . . 7 ⊢ 〈“𝑆”〉 = 〈“( I ‘𝑆)”〉 | |
17 | 16 | fveq1i 6696 | . . . . . 6 ⊢ (〈“𝑆”〉‘0) = (〈“( I ‘𝑆)”〉‘0) |
18 | fvex 6708 | . . . . . . 7 ⊢ ( I ‘𝑆) ∈ V | |
19 | s1fv 14132 | . . . . . . 7 ⊢ (( I ‘𝑆) ∈ V → (〈“( I ‘𝑆)”〉‘0) = ( I ‘𝑆)) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ (〈“( I ‘𝑆)”〉‘0) = ( I ‘𝑆) |
21 | 17, 20 | eqtri 2759 | . . . . 5 ⊢ (〈“𝑆”〉‘0) = ( I ‘𝑆) |
22 | 15, 21 | eqtri 2759 | . . . 4 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = ( I ‘𝑆) |
23 | 8, 22 | eqtri 2759 | . . 3 ⊢ ((reverse‘〈“𝑆”〉)‘0) = ( I ‘𝑆) |
24 | s1eq 14122 | . . 3 ⊢ (((reverse‘〈“𝑆”〉)‘0) = ( I ‘𝑆) → 〈“((reverse‘〈“𝑆”〉)‘0)”〉 = 〈“( I ‘𝑆)”〉) | |
25 | 23, 24 | ax-mp 5 | . 2 ⊢ 〈“((reverse‘〈“𝑆”〉)‘0)”〉 = 〈“( I ‘𝑆)”〉 |
26 | revcl 14291 | . . . 4 ⊢ (〈“𝑆”〉 ∈ Word V → (reverse‘〈“𝑆”〉) ∈ Word V) | |
27 | 1, 26 | ax-mp 5 | . . 3 ⊢ (reverse‘〈“𝑆”〉) ∈ Word V |
28 | revlen 14292 | . . . . 5 ⊢ (〈“𝑆”〉 ∈ Word V → (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉)) | |
29 | 1, 28 | ax-mp 5 | . . . 4 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉) |
30 | 29, 2 | eqtri 2759 | . . 3 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = 1 |
31 | eqs1 14134 | . . 3 ⊢ (((reverse‘〈“𝑆”〉) ∈ Word V ∧ (♯‘(reverse‘〈“𝑆”〉)) = 1) → (reverse‘〈“𝑆”〉) = 〈“((reverse‘〈“𝑆”〉)‘0)”〉) | |
32 | 27, 30, 31 | mp2an 692 | . 2 ⊢ (reverse‘〈“𝑆”〉) = 〈“((reverse‘〈“𝑆”〉)‘0)”〉 |
33 | 25, 32, 16 | 3eqtr4i 2769 | 1 ⊢ (reverse‘〈“𝑆”〉) = 〈“𝑆”〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 Vcvv 3398 I cid 5439 ‘cfv 6358 (class class class)co 7191 0cc0 10694 1c1 10695 − cmin 11027 ℕcn 11795 ..^cfzo 13203 ♯chash 13861 Word cword 14034 〈“cs1 14117 reversecreverse 14288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-s1 14118 df-reverse 14289 |
This theorem is referenced by: gsumwrev 18712 efginvrel2 19071 vrgpinv 19113 |
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