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Theorem repswrevw 14771

Description: The reverse of a "repeated symbol word". (Contributed by AV, 6-Nov-2018.)

**Assertion**
Ref	Expression
repswrevw	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (reverse‘(𝑆 repeatS 𝑁)) = (𝑆 repeatS 𝑁))

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

Step	Hyp	Ref	Expression
1		repswlen 14760	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (♯‘(𝑆 repeatS 𝑁)) = 𝑁)
2	1	oveq2d 7434	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (0..^(♯‘(𝑆 repeatS 𝑁))) = (0..^𝑁))
3	2	mpteq1d 5244	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑥 ∈ (0..^(♯‘(𝑆 repeatS 𝑁))) ↦ ((𝑆 repeatS 𝑁)‘(((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ ((𝑆 repeatS 𝑁)‘(((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥))))
4		simpll 765	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑆 ∈ 𝑉)
5		simplr 767	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ₀)
6	1	adantr 479	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑥 ∈ (0..^𝑁)) → (♯‘(𝑆 repeatS 𝑁)) = 𝑁)
7	6	oveq1d 7433	. . . . . . 7 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑥 ∈ (0..^𝑁)) → ((♯‘(𝑆 repeatS 𝑁)) − 1) = (𝑁 − 1))
8	7	oveq1d 7433	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑥 ∈ (0..^𝑁)) → (((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥) = ((𝑁 − 1) − 𝑥))
9		ubmelm1fzo 13762	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^𝑁) → ((𝑁 − 𝑥) − 1) ∈ (0..^𝑁))
10		elfzoelz 13665	. . . . . . . . 9 ⊢ (𝑥 ∈ (0..^𝑁) → 𝑥 ∈ ℤ)
11		nn0cn 12513	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
12	11	ad2antll 727	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑁 ∈ ℂ)
13		zcn 12594	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
14	13	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑥 ∈ ℂ)
15		1cnd 11240	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 1 ∈ ℂ)
16	12, 14, 15	sub32d 11634	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → ((𝑁 − 𝑥) − 1) = ((𝑁 − 1) − 𝑥))
17	16	eleq1d 2810	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (((𝑁 − 𝑥) − 1) ∈ (0..^𝑁) ↔ ((𝑁 − 1) − 𝑥) ∈ (0..^𝑁)))
18	17	biimpd 228	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (((𝑁 − 𝑥) − 1) ∈ (0..^𝑁) → ((𝑁 − 1) − 𝑥) ∈ (0..^𝑁)))
19	18	ex 411	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (((𝑁 − 𝑥) − 1) ∈ (0..^𝑁) → ((𝑁 − 1) − 𝑥) ∈ (0..^𝑁))))
20	10, 19	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^𝑁) → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (((𝑁 − 𝑥) − 1) ∈ (0..^𝑁) → ((𝑁 − 1) − 𝑥) ∈ (0..^𝑁))))
21	9, 20	mpid 44	. . . . . . 7 ⊢ (𝑥 ∈ (0..^𝑁) → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑁 − 1) − 𝑥) ∈ (0..^𝑁)))
22	21	impcom 406	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑁 − 1) − 𝑥) ∈ (0..^𝑁))
23	8, 22	eqeltrd 2825	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑥 ∈ (0..^𝑁)) → (((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥) ∈ (0..^𝑁))
24		repswsymb 14758	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀ ∧ (((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥) ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘(((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥)) = 𝑆)
25	4, 5, 23, 24	syl3anc 1368	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘(((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥)) = 𝑆)
26	25	mpteq2dva 5249	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑥 ∈ (0..^𝑁) ↦ ((𝑆 repeatS 𝑁)‘(((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
27	3, 26	eqtrd 2765	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑥 ∈ (0..^(♯‘(𝑆 repeatS 𝑁))) ↦ ((𝑆 repeatS 𝑁)‘(((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥))) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
28		ovex 7451	. . 3 ⊢ (𝑆 repeatS 𝑁) ∈ V
29		revval 14744	. . 3 ⊢ ((𝑆 repeatS 𝑁) ∈ V → (reverse‘(𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^(♯‘(𝑆 repeatS 𝑁))) ↦ ((𝑆 repeatS 𝑁)‘(((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥))))
30	28, 29	mp1i 13	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (reverse‘(𝑆 repeatS 𝑁)) = (𝑥 ∈ (0..^(♯‘(𝑆 repeatS 𝑁))) ↦ ((𝑆 repeatS 𝑁)‘(((♯‘(𝑆 repeatS 𝑁)) − 1) − 𝑥))))
31		reps 14754	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
32	27, 30, 31	3eqtr4d 2775	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (reverse‘(𝑆 repeatS 𝑁)) = (𝑆 repeatS 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ↦ cmpt 5232 ‘cfv 6548 (class class class)co 7418 ℂcc 11137 0cc0 11139 1c1 11140 − cmin 11475 ℕ₀cn0 12503 ℤcz 12589 ..^cfzo 13660 ♯chash 14323 reversecreverse 14742 repeatS creps 14752

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6306 df-ord 6373 df-on 6374 df-lim 6375 df-suc 6376 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7374 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14324 df-reverse 14743 df-reps 14753

This theorem is referenced by: (None)

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