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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 13962 | . . . . 5 ⊢ 〈“𝑆”〉 ∈ Word V | |
2 | s1len 13963 | . . . . . . 7 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 1nn 11652 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2, 3 | eqeltri 2912 | . . . . . 6 ⊢ (♯‘〈“𝑆”〉) ∈ ℕ |
5 | lbfzo0 13080 | . . . . . 6 ⊢ (0 ∈ (0..^(♯‘〈“𝑆”〉)) ↔ (♯‘〈“𝑆”〉) ∈ ℕ) | |
6 | 4, 5 | mpbir 233 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘〈“𝑆”〉)) |
7 | revfv 14128 | . . . . 5 ⊢ ((〈“𝑆”〉 ∈ Word V ∧ 0 ∈ (0..^(♯‘〈“𝑆”〉))) → ((reverse‘〈“𝑆”〉)‘0) = (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0))) | |
8 | 1, 6, 7 | mp2an 690 | . . . 4 ⊢ ((reverse‘〈“𝑆”〉)‘0) = (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) |
9 | 2 | oveq1i 7169 | . . . . . . . . 9 ⊢ ((♯‘〈“𝑆”〉) − 1) = (1 − 1) |
10 | 1m1e0 11712 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
11 | 9, 10 | eqtri 2847 | . . . . . . . 8 ⊢ ((♯‘〈“𝑆”〉) − 1) = 0 |
12 | 11 | oveq1i 7169 | . . . . . . 7 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = (0 − 0) |
13 | 0m0e0 11760 | . . . . . . 7 ⊢ (0 − 0) = 0 | |
14 | 12, 13 | eqtri 2847 | . . . . . 6 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = 0 |
15 | 14 | fveq2i 6676 | . . . . 5 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = (〈“𝑆”〉‘0) |
16 | ids1 13954 | . . . . . . 7 ⊢ 〈“𝑆”〉 = 〈“( I ‘𝑆)”〉 | |
17 | 16 | fveq1i 6674 | . . . . . 6 ⊢ (〈“𝑆”〉‘0) = (〈“( I ‘𝑆)”〉‘0) |
18 | fvex 6686 | . . . . . . 7 ⊢ ( I ‘𝑆) ∈ V | |
19 | s1fv 13967 | . . . . . . 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 2847 | . . . . 5 ⊢ (〈“𝑆”〉‘0) = ( I ‘𝑆) |
22 | 15, 21 | eqtri 2847 | . . . 4 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = ( I ‘𝑆) |
23 | 8, 22 | eqtri 2847 | . . 3 ⊢ ((reverse‘〈“𝑆”〉)‘0) = ( I ‘𝑆) |
24 | s1eq 13957 | . . 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 14126 | . . . 4 ⊢ (〈“𝑆”〉 ∈ Word V → (reverse‘〈“𝑆”〉) ∈ Word V) | |
27 | 1, 26 | ax-mp 5 | . . 3 ⊢ (reverse‘〈“𝑆”〉) ∈ Word V |
28 | revlen 14127 | . . . . 5 ⊢ (〈“𝑆”〉 ∈ Word V → (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉)) | |
29 | 1, 28 | ax-mp 5 | . . . 4 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉) |
30 | 29, 2 | eqtri 2847 | . . 3 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = 1 |
31 | eqs1 13969 | . . 3 ⊢ (((reverse‘〈“𝑆”〉) ∈ Word V ∧ (♯‘(reverse‘〈“𝑆”〉)) = 1) → (reverse‘〈“𝑆”〉) = 〈“((reverse‘〈“𝑆”〉)‘0)”〉) | |
32 | 27, 30, 31 | mp2an 690 | . 2 ⊢ (reverse‘〈“𝑆”〉) = 〈“((reverse‘〈“𝑆”〉)‘0)”〉 |
33 | 25, 32, 16 | 3eqtr4i 2857 | 1 ⊢ (reverse‘〈“𝑆”〉) = 〈“𝑆”〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 Vcvv 3497 I cid 5462 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 − cmin 10873 ℕcn 11641 ..^cfzo 13036 ♯chash 13693 Word cword 13864 〈“cs1 13952 reversecreverse 14123 |
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 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-s1 13953 df-reverse 14124 |
This theorem is referenced by: gsumwrev 18497 efginvrel2 18856 vrgpinv 18898 |
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