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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 14562 | . . . . 5 ⊢ 〈“𝑆”〉 ∈ Word V | |
2 | s1len 14563 | . . . . . . 7 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 1nn 12230 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 2, 3 | eqeltri 2828 | . . . . . 6 ⊢ (♯‘〈“𝑆”〉) ∈ ℕ |
5 | lbfzo0 13679 | . . . . . 6 ⊢ (0 ∈ (0..^(♯‘〈“𝑆”〉)) ↔ (♯‘〈“𝑆”〉) ∈ ℕ) | |
6 | 4, 5 | mpbir 230 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘〈“𝑆”〉)) |
7 | revfv 14720 | . . . . 5 ⊢ ((〈“𝑆”〉 ∈ Word V ∧ 0 ∈ (0..^(♯‘〈“𝑆”〉))) → ((reverse‘〈“𝑆”〉)‘0) = (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0))) | |
8 | 1, 6, 7 | mp2an 689 | . . . 4 ⊢ ((reverse‘〈“𝑆”〉)‘0) = (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) |
9 | 2 | oveq1i 7422 | . . . . . . . . 9 ⊢ ((♯‘〈“𝑆”〉) − 1) = (1 − 1) |
10 | 1m1e0 12291 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
11 | 9, 10 | eqtri 2759 | . . . . . . . 8 ⊢ ((♯‘〈“𝑆”〉) − 1) = 0 |
12 | 11 | oveq1i 7422 | . . . . . . 7 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = (0 − 0) |
13 | 0m0e0 12339 | . . . . . . 7 ⊢ (0 − 0) = 0 | |
14 | 12, 13 | eqtri 2759 | . . . . . 6 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = 0 |
15 | 14 | fveq2i 6894 | . . . . 5 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = (〈“𝑆”〉‘0) |
16 | ids1 14554 | . . . . . . 7 ⊢ 〈“𝑆”〉 = 〈“( I ‘𝑆)”〉 | |
17 | 16 | fveq1i 6892 | . . . . . 6 ⊢ (〈“𝑆”〉‘0) = (〈“( I ‘𝑆)”〉‘0) |
18 | fvex 6904 | . . . . . . 7 ⊢ ( I ‘𝑆) ∈ V | |
19 | s1fv 14567 | . . . . . . 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 14557 | . . 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 14718 | . . . 4 ⊢ (〈“𝑆”〉 ∈ Word V → (reverse‘〈“𝑆”〉) ∈ Word V) | |
27 | 1, 26 | ax-mp 5 | . . 3 ⊢ (reverse‘〈“𝑆”〉) ∈ Word V |
28 | revlen 14719 | . . . . 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 14569 | . . 3 ⊢ (((reverse‘〈“𝑆”〉) ∈ Word V ∧ (♯‘(reverse‘〈“𝑆”〉)) = 1) → (reverse‘〈“𝑆”〉) = 〈“((reverse‘〈“𝑆”〉)‘0)”〉) | |
32 | 27, 30, 31 | mp2an 689 | . 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 1540 ∈ wcel 2105 Vcvv 3473 I cid 5573 ‘cfv 6543 (class class class)co 7412 0cc0 11116 1c1 11117 − cmin 11451 ℕcn 12219 ..^cfzo 13634 ♯chash 14297 Word cword 14471 〈“cs1 14552 reversecreverse 14715 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-hash 14298 df-word 14472 df-s1 14553 df-reverse 14716 |
This theorem is referenced by: gsumwrev 19281 efginvrel2 19643 vrgpinv 19685 |
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