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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 13959 | . . . . 5 ⊢ 〈“𝑆”〉 ∈ Word V | |
2 | s1len 13960 | . . . . . . 7 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 1nn 11649 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2, 3 | eqeltri 2909 | . . . . . 6 ⊢ (♯‘〈“𝑆”〉) ∈ ℕ |
5 | lbfzo0 13078 | . . . . . 6 ⊢ (0 ∈ (0..^(♯‘〈“𝑆”〉)) ↔ (♯‘〈“𝑆”〉) ∈ ℕ) | |
6 | 4, 5 | mpbir 233 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘〈“𝑆”〉)) |
7 | revfv 14125 | . . . . 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 7166 | . . . . . . . . 9 ⊢ ((♯‘〈“𝑆”〉) − 1) = (1 − 1) |
10 | 1m1e0 11710 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
11 | 9, 10 | eqtri 2844 | . . . . . . . 8 ⊢ ((♯‘〈“𝑆”〉) − 1) = 0 |
12 | 11 | oveq1i 7166 | . . . . . . 7 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = (0 − 0) |
13 | 0m0e0 11758 | . . . . . . 7 ⊢ (0 − 0) = 0 | |
14 | 12, 13 | eqtri 2844 | . . . . . 6 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = 0 |
15 | 14 | fveq2i 6673 | . . . . 5 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = (〈“𝑆”〉‘0) |
16 | ids1 13951 | . . . . . . 7 ⊢ 〈“𝑆”〉 = 〈“( I ‘𝑆)”〉 | |
17 | 16 | fveq1i 6671 | . . . . . 6 ⊢ (〈“𝑆”〉‘0) = (〈“( I ‘𝑆)”〉‘0) |
18 | fvex 6683 | . . . . . . 7 ⊢ ( I ‘𝑆) ∈ V | |
19 | s1fv 13964 | . . . . . . 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 2844 | . . . . 5 ⊢ (〈“𝑆”〉‘0) = ( I ‘𝑆) |
22 | 15, 21 | eqtri 2844 | . . . 4 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = ( I ‘𝑆) |
23 | 8, 22 | eqtri 2844 | . . 3 ⊢ ((reverse‘〈“𝑆”〉)‘0) = ( I ‘𝑆) |
24 | s1eq 13954 | . . 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 14123 | . . . 4 ⊢ (〈“𝑆”〉 ∈ Word V → (reverse‘〈“𝑆”〉) ∈ Word V) | |
27 | 1, 26 | ax-mp 5 | . . 3 ⊢ (reverse‘〈“𝑆”〉) ∈ Word V |
28 | revlen 14124 | . . . . 5 ⊢ (〈“𝑆”〉 ∈ Word V → (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉)) | |
29 | 1, 28 | ax-mp 5 | . . . 4 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉) |
30 | 29, 2 | eqtri 2844 | . . 3 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = 1 |
31 | eqs1 13966 | . . 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 2854 | 1 ⊢ (reverse‘〈“𝑆”〉) = 〈“𝑆”〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 I cid 5459 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 − cmin 10870 ℕcn 11638 ..^cfzo 13034 ♯chash 13691 Word cword 13862 〈“cs1 13949 reversecreverse 14120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-s1 13950 df-reverse 14121 |
This theorem is referenced by: gsumwrev 18494 efginvrel2 18853 vrgpinv 18895 |
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