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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 14643 | . . . . 5 ⊢ 〈“𝑆”〉 ∈ Word V | |
| 2 | s1len 14644 | . . . . . . 7 ⊢ (♯‘〈“𝑆”〉) = 1 | |
| 3 | 1nn 12277 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
| 4 | 2, 3 | eqeltri 2837 | . . . . . 6 ⊢ (♯‘〈“𝑆”〉) ∈ ℕ |
| 5 | lbfzo0 13739 | . . . . . 6 ⊢ (0 ∈ (0..^(♯‘〈“𝑆”〉)) ↔ (♯‘〈“𝑆”〉) ∈ ℕ) | |
| 6 | 4, 5 | mpbir 231 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘〈“𝑆”〉)) |
| 7 | revfv 14801 | . . . . 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 7441 | . . . . . . . . 9 ⊢ ((♯‘〈“𝑆”〉) − 1) = (1 − 1) |
| 10 | 1m1e0 12338 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
| 11 | 9, 10 | eqtri 2765 | . . . . . . . 8 ⊢ ((♯‘〈“𝑆”〉) − 1) = 0 |
| 12 | 11 | oveq1i 7441 | . . . . . . 7 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = (0 − 0) |
| 13 | 0m0e0 12386 | . . . . . . 7 ⊢ (0 − 0) = 0 | |
| 14 | 12, 13 | eqtri 2765 | . . . . . 6 ⊢ (((♯‘〈“𝑆”〉) − 1) − 0) = 0 |
| 15 | 14 | fveq2i 6909 | . . . . 5 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = (〈“𝑆”〉‘0) |
| 16 | ids1 14635 | . . . . . . 7 ⊢ 〈“𝑆”〉 = 〈“( I ‘𝑆)”〉 | |
| 17 | 16 | fveq1i 6907 | . . . . . 6 ⊢ (〈“𝑆”〉‘0) = (〈“( I ‘𝑆)”〉‘0) |
| 18 | fvex 6919 | . . . . . . 7 ⊢ ( I ‘𝑆) ∈ V | |
| 19 | s1fv 14648 | . . . . . . 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 2765 | . . . . 5 ⊢ (〈“𝑆”〉‘0) = ( I ‘𝑆) |
| 22 | 15, 21 | eqtri 2765 | . . . 4 ⊢ (〈“𝑆”〉‘(((♯‘〈“𝑆”〉) − 1) − 0)) = ( I ‘𝑆) |
| 23 | 8, 22 | eqtri 2765 | . . 3 ⊢ ((reverse‘〈“𝑆”〉)‘0) = ( I ‘𝑆) |
| 24 | s1eq 14638 | . . 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 14799 | . . . 4 ⊢ (〈“𝑆”〉 ∈ Word V → (reverse‘〈“𝑆”〉) ∈ Word V) | |
| 27 | 1, 26 | ax-mp 5 | . . 3 ⊢ (reverse‘〈“𝑆”〉) ∈ Word V |
| 28 | revlen 14800 | . . . . 5 ⊢ (〈“𝑆”〉 ∈ Word V → (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉)) | |
| 29 | 1, 28 | ax-mp 5 | . . . 4 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = (♯‘〈“𝑆”〉) |
| 30 | 29, 2 | eqtri 2765 | . . 3 ⊢ (♯‘(reverse‘〈“𝑆”〉)) = 1 |
| 31 | eqs1 14650 | . . 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 2775 | 1 ⊢ (reverse‘〈“𝑆”〉) = 〈“𝑆”〉 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 I cid 5577 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 − cmin 11492 ℕcn 12266 ..^cfzo 13694 ♯chash 14369 Word cword 14552 〈“cs1 14633 reversecreverse 14796 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-s1 14634 df-reverse 14797 |
| This theorem is referenced by: gsumwrev 19385 efginvrel2 19745 vrgpinv 19787 |
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