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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 13766 | . . . . 5 ⊢ 〈“𝑆”〉 ∈ Word V | |
2 | s1len 13767 | . . . . . . 7 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 1nn 11450 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2, 3 | eqeltri 2856 | . . . . . 6 ⊢ (♯‘〈“𝑆”〉) ∈ ℕ |
5 | lbfzo0 12890 | . . . . . 6 ⊢ (0 ∈ (0..^(♯‘〈“𝑆”〉)) ↔ (♯‘〈“𝑆”〉) ∈ ℕ) | |
6 | 4, 5 | mpbir 223 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘〈“𝑆”〉)) |
7 | revfv 13980 | . . . . 5 ⊢ ((〈“𝑆”〉 ∈ Word V ∧ 0 ∈ (0..^(♯‘〈“𝑆”〉))) → ((reverse‘〈“𝑆”〉)‘0) = (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0))) | |
8 | 1, 6, 7 | mp2an 679 | . . . 4 ⊢ ((reverse‘〈“𝑆”〉)‘0) = (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) |
9 | 2 | oveq1i 6984 | . . . . . . . . 9 ⊢ ((♯‘〈“𝑆”〉) − 1) = (1 − 1) |
10 | 1m1e0 11510 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
11 | 9, 10 | eqtri 2796 | . . . . . . . 8 ⊢ ((♯‘〈“𝑆”〉) − 1) = 0 |
12 | 11 | oveq1i 6984 | . . . . . . 7 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = (0 − 0) |
13 | 0m0e0 11565 | . . . . . . 7 ⊢ (0 − 0) = 0 | |
14 | 12, 13 | eqtri 2796 | . . . . . 6 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = 0 |
15 | 14 | fveq2i 6499 | . . . . 5 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = (〈“𝑆”〉‘0) |
16 | ids1 13758 | . . . . . . 7 ⊢ 〈“𝑆”〉 = 〈“( I ‘𝑆)”〉 | |
17 | 16 | fveq1i 6497 | . . . . . 6 ⊢ (〈“𝑆”〉‘0) = (〈“( I ‘𝑆)”〉‘0) |
18 | fvex 6509 | . . . . . . 7 ⊢ ( I ‘𝑆) ∈ V | |
19 | s1fv 13771 | . . . . . . 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 2796 | . . . . 5 ⊢ (〈“𝑆”〉‘0) = ( I ‘𝑆) |
22 | 15, 21 | eqtri 2796 | . . . 4 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = ( I ‘𝑆) |
23 | 8, 22 | eqtri 2796 | . . 3 ⊢ ((reverse‘〈“𝑆”〉)‘0) = ( I ‘𝑆) |
24 | s1eq 13761 | . . 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 13978 | . . . 4 ⊢ (〈“𝑆”〉 ∈ Word V → (reverse‘〈“𝑆”〉) ∈ Word V) | |
27 | 1, 26 | ax-mp 5 | . . 3 ⊢ (reverse‘〈“𝑆”〉) ∈ Word V |
28 | revlen 13979 | . . . . 5 ⊢ (〈“𝑆”〉 ∈ Word V → (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉)) | |
29 | 1, 28 | ax-mp 5 | . . . 4 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉) |
30 | 29, 2 | eqtri 2796 | . . 3 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = 1 |
31 | eqs1 13773 | . . 3 ⊢ (((reverse‘〈“𝑆”〉) ∈ Word V ∧ (♯‘(reverse‘〈“𝑆”〉)) = 1) → (reverse‘〈“𝑆”〉) = 〈“((reverse‘〈“𝑆”〉)‘0)”〉) | |
32 | 27, 30, 31 | mp2an 679 | . 2 ⊢ (reverse‘〈“𝑆”〉) = 〈“((reverse‘〈“𝑆”〉)‘0)”〉 |
33 | 25, 32, 16 | 3eqtr4i 2806 | 1 ⊢ (reverse‘〈“𝑆”〉) = 〈“𝑆”〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 Vcvv 3409 I cid 5307 ‘cfv 6185 (class class class)co 6974 0cc0 10333 1c1 10334 − cmin 10668 ℕcn 11437 ..^cfzo 12847 ♯chash 13503 Word cword 13670 〈“cs1 13756 reversecreverse 13975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-fzo 12848 df-hash 13504 df-word 13671 df-s1 13757 df-reverse 13976 |
This theorem is referenced by: gsumwrev 18277 efginvrel2 18623 vrgpinv 18667 |
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