revrev - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > revrev		Structured version Visualization version GIF version

Theorem revrev 14717

Description: Reversal is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)

**Assertion**
Ref	Expression
revrev	⊢ (𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) = 𝑊)

Proof of Theorem revrev Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		revcl 14711	. . . 4 ⊢ (𝑊 ∈ Word 𝐴 → (reverse‘𝑊) ∈ Word 𝐴)
2		revcl 14711	. . . 4 ⊢ ((reverse‘𝑊) ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) ∈ Word 𝐴)
3		wrdf 14469	. . . 4 ⊢ ((reverse‘(reverse‘𝑊)) ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)):(0..^(♯‘(reverse‘(reverse‘𝑊))))⟶𝐴)
4		ffn 6718	. . . 4 ⊢ ((reverse‘(reverse‘𝑊)):(0..^(♯‘(reverse‘(reverse‘𝑊))))⟶𝐴 → (reverse‘(reverse‘𝑊)) Fn (0..^(♯‘(reverse‘(reverse‘𝑊)))))
5	1, 2, 3, 4	4syl 19	. . 3 ⊢ (𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) Fn (0..^(♯‘(reverse‘(reverse‘𝑊)))))
6		revlen 14712	. . . . . . 7 ⊢ ((reverse‘𝑊) ∈ Word 𝐴 → (♯‘(reverse‘(reverse‘𝑊))) = (♯‘(reverse‘𝑊)))
7	1, 6	syl 17	. . . . . 6 ⊢ (𝑊 ∈ Word 𝐴 → (♯‘(reverse‘(reverse‘𝑊))) = (♯‘(reverse‘𝑊)))
8		revlen 14712	. . . . . 6 ⊢ (𝑊 ∈ Word 𝐴 → (♯‘(reverse‘𝑊)) = (♯‘𝑊))
9	7, 8	eqtrd 2773	. . . . 5 ⊢ (𝑊 ∈ Word 𝐴 → (♯‘(reverse‘(reverse‘𝑊))) = (♯‘𝑊))
10	9	oveq2d 7425	. . . 4 ⊢ (𝑊 ∈ Word 𝐴 → (0..^(♯‘(reverse‘(reverse‘𝑊)))) = (0..^(♯‘𝑊)))
11	10	fneq2d 6644	. . 3 ⊢ (𝑊 ∈ Word 𝐴 → ((reverse‘(reverse‘𝑊)) Fn (0..^(♯‘(reverse‘(reverse‘𝑊)))) ↔ (reverse‘(reverse‘𝑊)) Fn (0..^(♯‘𝑊))))
12	5, 11	mpbid 231	. 2 ⊢ (𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) Fn (0..^(♯‘𝑊)))
13		wrdfn 14478	. 2 ⊢ (𝑊 ∈ Word 𝐴 → 𝑊 Fn (0..^(♯‘𝑊)))
14		simpr 486	. . . . 5 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → 𝑥 ∈ (0..^(♯‘𝑊)))
15	8	adantr 482	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (♯‘(reverse‘𝑊)) = (♯‘𝑊))
16	15	oveq2d 7425	. . . . 5 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (0..^(♯‘(reverse‘𝑊))) = (0..^(♯‘𝑊)))
17	14, 16	eleqtrrd 2837	. . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → 𝑥 ∈ (0..^(♯‘(reverse‘𝑊))))
18		revfv 14713	. . . 4 ⊢ (((reverse‘𝑊) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(reverse‘𝑊)))) → ((reverse‘(reverse‘𝑊))‘𝑥) = ((reverse‘𝑊)‘(((♯‘(reverse‘𝑊)) − 1) − 𝑥)))
19	1, 17, 18	syl2an2r 684	. . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((reverse‘(reverse‘𝑊))‘𝑥) = ((reverse‘𝑊)‘(((♯‘(reverse‘𝑊)) − 1) − 𝑥)))
20	15	oveq1d 7424	. . . . 5 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((♯‘(reverse‘𝑊)) − 1) = ((♯‘𝑊) − 1))
21	20	fvoveq1d 7431	. . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((reverse‘𝑊)‘(((♯‘(reverse‘𝑊)) − 1) − 𝑥)) = ((reverse‘𝑊)‘(((♯‘𝑊) − 1) − 𝑥)))
22		lencl 14483	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝐴 → (♯‘𝑊) ∈ ℕ₀)
23	22	nn0zd 12584	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐴 → (♯‘𝑊) ∈ ℤ)
24		fzoval 13633	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) ∈ ℤ → (0..^(♯‘𝑊)) = (0...((♯‘𝑊) − 1)))
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝐴 → (0..^(♯‘𝑊)) = (0...((♯‘𝑊) − 1)))
26	25	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝐴 → (𝑥 ∈ (0..^(♯‘𝑊)) ↔ 𝑥 ∈ (0...((♯‘𝑊) − 1))))
27	26	biimpa 478	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → 𝑥 ∈ (0...((♯‘𝑊) − 1)))
28		fznn0sub2 13608	. . . . . . . 8 ⊢ (𝑥 ∈ (0...((♯‘𝑊) − 1)) → (((♯‘𝑊) − 1) − 𝑥) ∈ (0...((♯‘𝑊) − 1)))
29	27, 28	syl 17	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (((♯‘𝑊) − 1) − 𝑥) ∈ (0...((♯‘𝑊) − 1)))
30	25	adantr 482	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (0..^(♯‘𝑊)) = (0...((♯‘𝑊) − 1)))
31	29, 30	eleqtrrd 2837	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (((♯‘𝑊) − 1) − 𝑥) ∈ (0..^(♯‘𝑊)))
32		revfv 14713	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝐴 ∧ (((♯‘𝑊) − 1) − 𝑥) ∈ (0..^(♯‘𝑊))) → ((reverse‘𝑊)‘(((♯‘𝑊) − 1) − 𝑥)) = (𝑊‘(((♯‘𝑊) − 1) − (((♯‘𝑊) − 1) − 𝑥))))
33	31, 32	syldan 592	. . . . 5 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((reverse‘𝑊)‘(((♯‘𝑊) − 1) − 𝑥)) = (𝑊‘(((♯‘𝑊) − 1) − (((♯‘𝑊) − 1) − 𝑥))))
34		peano2zm 12605	. . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ ℤ → ((♯‘𝑊) − 1) ∈ ℤ)
35	23, 34	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝐴 → ((♯‘𝑊) − 1) ∈ ℤ)
36	35	zcnd 12667	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝐴 → ((♯‘𝑊) − 1) ∈ ℂ)
37		elfzoelz 13632	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^(♯‘𝑊)) → 𝑥 ∈ ℤ)
38	37	zcnd 12667	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(♯‘𝑊)) → 𝑥 ∈ ℂ)
39		nncan 11489	. . . . . . 7 ⊢ ((((♯‘𝑊) − 1) ∈ ℂ ∧ 𝑥 ∈ ℂ) → (((♯‘𝑊) − 1) − (((♯‘𝑊) − 1) − 𝑥)) = 𝑥)
40	36, 38, 39	syl2an 597	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (((♯‘𝑊) − 1) − (((♯‘𝑊) − 1) − 𝑥)) = 𝑥)
41	40	fveq2d 6896	. . . . 5 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (𝑊‘(((♯‘𝑊) − 1) − (((♯‘𝑊) − 1) − 𝑥))) = (𝑊‘𝑥))
42	33, 41	eqtrd 2773	. . . 4 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((reverse‘𝑊)‘(((♯‘𝑊) − 1) − 𝑥)) = (𝑊‘𝑥))
43	21, 42	eqtrd 2773	. . 3 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((reverse‘𝑊)‘(((♯‘(reverse‘𝑊)) − 1) − 𝑥)) = (𝑊‘𝑥))
44	19, 43	eqtrd 2773	. 2 ⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((reverse‘(reverse‘𝑊))‘𝑥) = (𝑊‘𝑥))
45	12, 13, 44	eqfnfvd 7036	1 ⊢ (𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) = 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 0cc0 11110 1c1 11111 − cmin 11444 ℤcz 12558 ...cfz 13484 ..^cfzo 13627 ♯chash 14290 Word cword 14464 reversecreverse 14708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-reverse 14709

This theorem is referenced by: efginvrel1 19596 swrdrevpfx 34107 revwlkb 34116

W3C validator