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Theorem revccat 14661
Description: Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Assertion
Ref Expression
revccat ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (reverseβ€˜(𝑆 ++ 𝑇)) = ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)))

Proof of Theorem revccat
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ccatcl 14469 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (𝑆 ++ 𝑇) ∈ Word 𝐴)
2 revcl 14656 . . . 4 ((𝑆 ++ 𝑇) ∈ Word 𝐴 β†’ (reverseβ€˜(𝑆 ++ 𝑇)) ∈ Word 𝐴)
3 wrdfn 14423 . . . 4 ((reverseβ€˜(𝑆 ++ 𝑇)) ∈ Word 𝐴 β†’ (reverseβ€˜(𝑆 ++ 𝑇)) Fn (0..^(β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇)))))
41, 2, 33syl 18 . . 3 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (reverseβ€˜(𝑆 ++ 𝑇)) Fn (0..^(β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇)))))
5 revlen 14657 . . . . . . 7 ((𝑆 ++ 𝑇) ∈ Word 𝐴 β†’ (β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇))) = (β™―β€˜(𝑆 ++ 𝑇)))
61, 5syl 17 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇))) = (β™―β€˜(𝑆 ++ 𝑇)))
7 ccatlen 14470 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(𝑆 ++ 𝑇)) = ((β™―β€˜π‘†) + (β™―β€˜π‘‡)))
8 lencl 14428 . . . . . . . 8 (𝑆 ∈ Word 𝐴 β†’ (β™―β€˜π‘†) ∈ β„•0)
98nn0cnd 12482 . . . . . . 7 (𝑆 ∈ Word 𝐴 β†’ (β™―β€˜π‘†) ∈ β„‚)
10 lencl 14428 . . . . . . . 8 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜π‘‡) ∈ β„•0)
1110nn0cnd 12482 . . . . . . 7 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜π‘‡) ∈ β„‚)
12 addcom 11348 . . . . . . 7 (((β™―β€˜π‘†) ∈ β„‚ ∧ (β™―β€˜π‘‡) ∈ β„‚) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
139, 11, 12syl2an 597 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
146, 7, 133eqtrd 2781 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇))) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
1514oveq2d 7378 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇)))) = (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
1615fneq2d 6601 . . 3 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((reverseβ€˜(𝑆 ++ 𝑇)) Fn (0..^(β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇)))) ↔ (reverseβ€˜(𝑆 ++ 𝑇)) Fn (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))))
174, 16mpbid 231 . 2 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (reverseβ€˜(𝑆 ++ 𝑇)) Fn (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
18 revcl 14656 . . . . 5 (𝑇 ∈ Word 𝐴 β†’ (reverseβ€˜π‘‡) ∈ Word 𝐴)
19 revcl 14656 . . . . 5 (𝑆 ∈ Word 𝐴 β†’ (reverseβ€˜π‘†) ∈ Word 𝐴)
20 ccatcl 14469 . . . . 5 (((reverseβ€˜π‘‡) ∈ Word 𝐴 ∧ (reverseβ€˜π‘†) ∈ Word 𝐴) β†’ ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) ∈ Word 𝐴)
2118, 19, 20syl2anr 598 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) ∈ Word 𝐴)
22 wrdfn 14423 . . . 4 (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) ∈ Word 𝐴 β†’ ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) Fn (0..^(β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)))))
2321, 22syl 17 . . 3 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) Fn (0..^(β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)))))
24 ccatlen 14470 . . . . . . 7 (((reverseβ€˜π‘‡) ∈ Word 𝐴 ∧ (reverseβ€˜π‘†) ∈ Word 𝐴) β†’ (β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))) = ((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†))))
2518, 19, 24syl2anr 598 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))) = ((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†))))
26 revlen 14657 . . . . . . 7 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜(reverseβ€˜π‘‡)) = (β™―β€˜π‘‡))
27 revlen 14657 . . . . . . 7 (𝑆 ∈ Word 𝐴 β†’ (β™―β€˜(reverseβ€˜π‘†)) = (β™―β€˜π‘†))
2826, 27oveqan12rd 7382 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†))) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
2925, 28eqtrd 2777 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
3029oveq2d 7378 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)))) = (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
3130fneq2d 6601 . . 3 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) Fn (0..^(β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)))) ↔ ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) Fn (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))))
3223, 31mpbid 231 . 2 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) Fn (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
33 id 22 . . . 4 (π‘₯ ∈ (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) β†’ π‘₯ ∈ (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
3410nn0zd 12532 . . . . 5 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜π‘‡) ∈ β„€)
3534adantl 483 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜π‘‡) ∈ β„€)
36 fzospliti 13611 . . . 4 ((π‘₯ ∈ (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) ∧ (β™―β€˜π‘‡) ∈ β„€) β†’ (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) ∨ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))))
3733, 35, 36syl2anr 598 . . 3 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) ∨ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))))
38 simpll 766 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ 𝑆 ∈ Word 𝐴)
39 simplr 768 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ 𝑇 ∈ Word 𝐴)
40 fzoval 13580 . . . . . . . . . . . . 13 ((β™―β€˜π‘‡) ∈ β„€ β†’ (0..^(β™―β€˜π‘‡)) = (0...((β™―β€˜π‘‡) βˆ’ 1)))
4134, 40syl 17 . . . . . . . . . . . 12 (𝑇 ∈ Word 𝐴 β†’ (0..^(β™―β€˜π‘‡)) = (0...((β™―β€˜π‘‡) βˆ’ 1)))
4241adantl 483 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜π‘‡)) = (0...((β™―β€˜π‘‡) βˆ’ 1)))
4342eleq2d 2824 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) ↔ π‘₯ ∈ (0...((β™―β€˜π‘‡) βˆ’ 1))))
4443biimpa 478 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ (0...((β™―β€˜π‘‡) βˆ’ 1)))
45 fznn0sub2 13555 . . . . . . . . 9 (π‘₯ ∈ (0...((β™―β€˜π‘‡) βˆ’ 1)) β†’ (((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) ∈ (0...((β™―β€˜π‘‡) βˆ’ 1)))
4644, 45syl 17 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) ∈ (0...((β™―β€˜π‘‡) βˆ’ 1)))
4741ad2antlr 726 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (0..^(β™―β€˜π‘‡)) = (0...((β™―β€˜π‘‡) βˆ’ 1)))
4846, 47eleqtrrd 2841 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) ∈ (0..^(β™―β€˜π‘‡)))
49 ccatval3 14474 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ (((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜((((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) + (β™―β€˜π‘†))) = (π‘‡β€˜(((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯)))
5038, 39, 48, 49syl3anc 1372 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜((((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) + (β™―β€˜π‘†))) = (π‘‡β€˜(((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯)))
517, 13eqtrd 2777 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(𝑆 ++ 𝑇)) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
5251oveq1d 7377 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) = (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))
5311adantl 483 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜π‘‡) ∈ β„‚)
549adantr 482 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜π‘†) ∈ β„‚)
55 1cnd 11157 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ 1 ∈ β„‚)
5653, 54, 55addsubd 11540 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) = (((β™―β€˜π‘‡) βˆ’ 1) + (β™―β€˜π‘†)))
5752, 56eqtrd 2777 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) = (((β™―β€˜π‘‡) βˆ’ 1) + (β™―β€˜π‘†)))
5857oveq1d 7377 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) = ((((β™―β€˜π‘‡) βˆ’ 1) + (β™―β€˜π‘†)) βˆ’ π‘₯))
5958adantr 482 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) = ((((β™―β€˜π‘‡) βˆ’ 1) + (β™―β€˜π‘†)) βˆ’ π‘₯))
60 peano2zm 12553 . . . . . . . . . . . 12 ((β™―β€˜π‘‡) ∈ β„€ β†’ ((β™―β€˜π‘‡) βˆ’ 1) ∈ β„€)
6134, 60syl 17 . . . . . . . . . . 11 (𝑇 ∈ Word 𝐴 β†’ ((β™―β€˜π‘‡) βˆ’ 1) ∈ β„€)
6261zcnd 12615 . . . . . . . . . 10 (𝑇 ∈ Word 𝐴 β†’ ((β™―β€˜π‘‡) βˆ’ 1) ∈ β„‚)
6362ad2antlr 726 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((β™―β€˜π‘‡) βˆ’ 1) ∈ β„‚)
649ad2antrr 725 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (β™―β€˜π‘†) ∈ β„‚)
65 elfzoelz 13579 . . . . . . . . . . 11 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ ∈ β„€)
6665zcnd 12615 . . . . . . . . . 10 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ ∈ β„‚)
6766adantl 483 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ β„‚)
6863, 64, 67addsubd 11540 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((((β™―β€˜π‘‡) βˆ’ 1) + (β™―β€˜π‘†)) βˆ’ π‘₯) = ((((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) + (β™―β€˜π‘†)))
6959, 68eqtrd 2777 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) = ((((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) + (β™―β€˜π‘†)))
7069fveq2d 6851 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)) = ((𝑆 ++ 𝑇)β€˜((((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯) + (β™―β€˜π‘†))))
71 revfv 14658 . . . . . . 7 ((𝑇 ∈ Word 𝐴 ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((reverseβ€˜π‘‡)β€˜π‘₯) = (π‘‡β€˜(((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯)))
7271adantll 713 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((reverseβ€˜π‘‡)β€˜π‘₯) = (π‘‡β€˜(((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯)))
7350, 70, 723eqtr4d 2787 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)) = ((reverseβ€˜π‘‡)β€˜π‘₯))
7434uzidd 12786 . . . . . . . . . 10 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜π‘‡) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)))
75 uzaddcl 12836 . . . . . . . . . 10 (((β™―β€˜π‘‡) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)) ∧ (β™―β€˜π‘†) ∈ β„•0) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)))
7674, 8, 75syl2anr 598 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)))
7751, 76eqeltrd 2838 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(𝑆 ++ 𝑇)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)))
78 fzoss2 13607 . . . . . . . 8 ((β™―β€˜(𝑆 ++ 𝑇)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)) β†’ (0..^(β™―β€˜π‘‡)) βŠ† (0..^(β™―β€˜(𝑆 ++ 𝑇))))
7977, 78syl 17 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜π‘‡)) βŠ† (0..^(β™―β€˜(𝑆 ++ 𝑇))))
8079sselda 3949 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ (0..^(β™―β€˜(𝑆 ++ 𝑇))))
81 revfv 14658 . . . . . 6 (((𝑆 ++ 𝑇) ∈ Word 𝐴 ∧ π‘₯ ∈ (0..^(β™―β€˜(𝑆 ++ 𝑇)))) β†’ ((reverseβ€˜(𝑆 ++ 𝑇))β€˜π‘₯) = ((𝑆 ++ 𝑇)β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)))
821, 80, 81syl2an2r 684 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((reverseβ€˜(𝑆 ++ 𝑇))β€˜π‘₯) = ((𝑆 ++ 𝑇)β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)))
8318ad2antlr 726 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (reverseβ€˜π‘‡) ∈ Word 𝐴)
8419ad2antrr 725 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (reverseβ€˜π‘†) ∈ Word 𝐴)
8526adantl 483 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(reverseβ€˜π‘‡)) = (β™―β€˜π‘‡))
8685oveq2d 7378 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜(reverseβ€˜π‘‡))) = (0..^(β™―β€˜π‘‡)))
8786eleq2d 2824 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (π‘₯ ∈ (0..^(β™―β€˜(reverseβ€˜π‘‡))) ↔ π‘₯ ∈ (0..^(β™―β€˜π‘‡))))
8887biimpar 479 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ (0..^(β™―β€˜(reverseβ€˜π‘‡))))
89 ccatval1 14472 . . . . . 6 (((reverseβ€˜π‘‡) ∈ Word 𝐴 ∧ (reverseβ€˜π‘†) ∈ Word 𝐴 ∧ π‘₯ ∈ (0..^(β™―β€˜(reverseβ€˜π‘‡)))) β†’ (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))β€˜π‘₯) = ((reverseβ€˜π‘‡)β€˜π‘₯))
9083, 84, 88, 89syl3anc 1372 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))β€˜π‘₯) = ((reverseβ€˜π‘‡)β€˜π‘₯))
9173, 82, 903eqtr4d 2787 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((reverseβ€˜(𝑆 ++ 𝑇))β€˜π‘₯) = (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))β€˜π‘₯))
928nn0zd 12532 . . . . . . . . . . . 12 (𝑆 ∈ Word 𝐴 β†’ (β™―β€˜π‘†) ∈ β„€)
93 peano2zm 12553 . . . . . . . . . . . 12 ((β™―β€˜π‘†) ∈ β„€ β†’ ((β™―β€˜π‘†) βˆ’ 1) ∈ β„€)
9492, 93syl 17 . . . . . . . . . . 11 (𝑆 ∈ Word 𝐴 β†’ ((β™―β€˜π‘†) βˆ’ 1) ∈ β„€)
9594zcnd 12615 . . . . . . . . . 10 (𝑆 ∈ Word 𝐴 β†’ ((β™―β€˜π‘†) βˆ’ 1) ∈ β„‚)
9695ad2antrr 725 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ ((β™―β€˜π‘†) βˆ’ 1) ∈ β„‚)
97 elfzoelz 13579 . . . . . . . . . . 11 (π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) β†’ π‘₯ ∈ β„€)
9897zcnd 12615 . . . . . . . . . 10 (π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) β†’ π‘₯ ∈ β„‚)
9998adantl 483 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ π‘₯ ∈ β„‚)
10011ad2antlr 726 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (β™―β€˜π‘‡) ∈ β„‚)
10196, 99, 100subsub3d 11549 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜π‘‡))) = ((((β™―β€˜π‘†) βˆ’ 1) + (β™―β€˜π‘‡)) βˆ’ π‘₯))
10226oveq2d 7378 . . . . . . . . . 10 (𝑇 ∈ Word 𝐴 β†’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡))) = (π‘₯ βˆ’ (β™―β€˜π‘‡)))
103102oveq2d 7378 . . . . . . . . 9 (𝑇 ∈ Word 𝐴 β†’ (((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡)))) = (((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜π‘‡))))
104103ad2antlr 726 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡)))) = (((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜π‘‡))))
1057oveq1d 7377 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) = (((β™―β€˜π‘†) + (β™―β€˜π‘‡)) βˆ’ 1))
10654, 53, 55addsubd 11540 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜π‘†) + (β™―β€˜π‘‡)) βˆ’ 1) = (((β™―β€˜π‘†) βˆ’ 1) + (β™―β€˜π‘‡)))
107105, 106eqtrd 2777 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) = (((β™―β€˜π‘†) βˆ’ 1) + (β™―β€˜π‘‡)))
108107oveq1d 7377 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) = ((((β™―β€˜π‘†) βˆ’ 1) + (β™―β€˜π‘‡)) βˆ’ π‘₯))
109108adantr 482 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) = ((((β™―β€˜π‘†) βˆ’ 1) + (β™―β€˜π‘‡)) βˆ’ π‘₯))
110101, 104, 1093eqtr4rd 2788 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) = (((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡)))))
111110fveq2d 6851 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (π‘†β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)) = (π‘†β€˜(((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡))))))
112 simpll 766 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ 𝑆 ∈ Word 𝐴)
113 simplr 768 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ 𝑇 ∈ Word 𝐴)
114 zaddcl 12550 . . . . . . . . . . 11 (((β™―β€˜π‘‡) ∈ β„€ ∧ (β™―β€˜π‘†) ∈ β„€) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„€)
11534, 92, 114syl2anr 598 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„€)
116 peano2zm 12553 . . . . . . . . . 10 (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„€ β†’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) ∈ β„€)
117115, 116syl 17 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) ∈ β„€)
118 fzoval 13580 . . . . . . . . . . . 12 (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„€ β†’ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) = ((β™―β€˜π‘‡)...(((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1)))
119115, 118syl 17 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) = ((β™―β€˜π‘‡)...(((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1)))
120119eleq2d 2824 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) ↔ π‘₯ ∈ ((β™―β€˜π‘‡)...(((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))))
121120biimpa 478 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ π‘₯ ∈ ((β™―β€˜π‘‡)...(((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1)))
122 fzrev2i 13513 . . . . . . . . 9 (((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) ∈ β„€ ∧ π‘₯ ∈ ((β™―β€˜π‘‡)...(((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))) β†’ ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ π‘₯) ∈ (((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))...((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (β™―β€˜π‘‡))))
123117, 121, 122syl2an2r 684 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ π‘₯) ∈ (((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))...((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (β™―β€˜π‘‡))))
12452oveq1d 7377 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) = ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ π‘₯))
125124adantr 482 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) = ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ π‘₯))
12692adantr 482 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜π‘†) ∈ β„€)
127 fzoval 13580 . . . . . . . . . . 11 ((β™―β€˜π‘†) ∈ β„€ β†’ (0..^(β™―β€˜π‘†)) = (0...((β™―β€˜π‘†) βˆ’ 1)))
128126, 127syl 17 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜π‘†)) = (0...((β™―β€˜π‘†) βˆ’ 1)))
129117zcnd 12615 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) ∈ β„‚)
130129subidd 11507 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1)) = 0)
131 addcl 11140 . . . . . . . . . . . . . 14 (((β™―β€˜π‘‡) ∈ β„‚ ∧ (β™―β€˜π‘†) ∈ β„‚) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„‚)
13211, 9, 131syl2anr 598 . . . . . . . . . . . . 13 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„‚)
133132, 55, 53sub32d 11551 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (β™―β€˜π‘‡)) = ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ (β™―β€˜π‘‡)) βˆ’ 1))
134 pncan2 11415 . . . . . . . . . . . . . 14 (((β™―β€˜π‘‡) ∈ β„‚ ∧ (β™―β€˜π‘†) ∈ β„‚) β†’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ (β™―β€˜π‘‡)) = (β™―β€˜π‘†))
13511, 9, 134syl2anr 598 . . . . . . . . . . . . 13 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ (β™―β€˜π‘‡)) = (β™―β€˜π‘†))
136135oveq1d 7377 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ (β™―β€˜π‘‡)) βˆ’ 1) = ((β™―β€˜π‘†) βˆ’ 1))
137133, 136eqtrd 2777 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (β™―β€˜π‘‡)) = ((β™―β€˜π‘†) βˆ’ 1))
138130, 137oveq12d 7380 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))...((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (β™―β€˜π‘‡))) = (0...((β™―β€˜π‘†) βˆ’ 1)))
139128, 138eqtr4d 2780 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜π‘†)) = (((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))...((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (β™―β€˜π‘‡))))
140139adantr 482 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (0..^(β™―β€˜π‘†)) = (((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))...((((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) βˆ’ (β™―β€˜π‘‡))))
141123, 125, 1403eltr4d 2853 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) ∈ (0..^(β™―β€˜π‘†)))
142 ccatval1 14472 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ (((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯) ∈ (0..^(β™―β€˜π‘†))) β†’ ((𝑆 ++ 𝑇)β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)) = (π‘†β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)))
143112, 113, 141, 142syl3anc 1372 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ ((𝑆 ++ 𝑇)β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)) = (π‘†β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)))
144 simpl 484 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ 𝑆 ∈ Word 𝐴)
145102ad2antlr 726 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡))) = (π‘₯ βˆ’ (β™―β€˜π‘‡)))
146 id 22 . . . . . . . . 9 (π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) β†’ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
147 fzosubel3 13640 . . . . . . . . 9 ((π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) ∧ (β™―β€˜π‘†) ∈ β„€) β†’ (π‘₯ βˆ’ (β™―β€˜π‘‡)) ∈ (0..^(β™―β€˜π‘†)))
148146, 126, 147syl2anr 598 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (π‘₯ βˆ’ (β™―β€˜π‘‡)) ∈ (0..^(β™―β€˜π‘†)))
149145, 148eqeltrd 2838 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡))) ∈ (0..^(β™―β€˜π‘†)))
150 revfv 14658 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡))) ∈ (0..^(β™―β€˜π‘†))) β†’ ((reverseβ€˜π‘†)β€˜(π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡)))) = (π‘†β€˜(((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡))))))
151144, 149, 150syl2an2r 684 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ ((reverseβ€˜π‘†)β€˜(π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡)))) = (π‘†β€˜(((β™―β€˜π‘†) βˆ’ 1) βˆ’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡))))))
152111, 143, 1513eqtr4d 2787 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ ((𝑆 ++ 𝑇)β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)) = ((reverseβ€˜π‘†)β€˜(π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡)))))
153 fzoss1 13606 . . . . . . . . . . 11 ((β™―β€˜π‘‡) ∈ (β„€β‰₯β€˜0) β†’ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) βŠ† (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
154 nn0uz 12812 . . . . . . . . . . 11 β„•0 = (β„€β‰₯β€˜0)
155153, 154eleq2s 2856 . . . . . . . . . 10 ((β™―β€˜π‘‡) ∈ β„•0 β†’ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) βŠ† (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
15610, 155syl 17 . . . . . . . . 9 (𝑇 ∈ Word 𝐴 β†’ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) βŠ† (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
157156adantl 483 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) βŠ† (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
15851oveq2d 7378 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜(𝑆 ++ 𝑇))) = (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
159157, 158sseqtrrd 3990 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) βŠ† (0..^(β™―β€˜(𝑆 ++ 𝑇))))
160159sselda 3949 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ π‘₯ ∈ (0..^(β™―β€˜(𝑆 ++ 𝑇))))
1611, 160, 81syl2an2r 684 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ ((reverseβ€˜(𝑆 ++ 𝑇))β€˜π‘₯) = ((𝑆 ++ 𝑇)β€˜(((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) βˆ’ π‘₯)))
16218ad2antlr 726 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (reverseβ€˜π‘‡) ∈ Word 𝐴)
16319ad2antrr 725 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (reverseβ€˜π‘†) ∈ Word 𝐴)
16485, 28oveq12d 7380 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(reverseβ€˜π‘‡))..^((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†)))) = ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
165164eleq2d 2824 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (π‘₯ ∈ ((β™―β€˜(reverseβ€˜π‘‡))..^((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†)))) ↔ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))))
166165biimpar 479 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ π‘₯ ∈ ((β™―β€˜(reverseβ€˜π‘‡))..^((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†)))))
167 ccatval2 14473 . . . . . 6 (((reverseβ€˜π‘‡) ∈ Word 𝐴 ∧ (reverseβ€˜π‘†) ∈ Word 𝐴 ∧ π‘₯ ∈ ((β™―β€˜(reverseβ€˜π‘‡))..^((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†))))) β†’ (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))β€˜π‘₯) = ((reverseβ€˜π‘†)β€˜(π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡)))))
168162, 163, 166, 167syl3anc 1372 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))β€˜π‘₯) = ((reverseβ€˜π‘†)β€˜(π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡)))))
169152, 161, 1683eqtr4d 2787 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ ((reverseβ€˜(𝑆 ++ 𝑇))β€˜π‘₯) = (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))β€˜π‘₯))
17091, 169jaodan 957 . . 3 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) ∨ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))) β†’ ((reverseβ€˜(𝑆 ++ 𝑇))β€˜π‘₯) = (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))β€˜π‘₯))
17137, 170syldan 592 . 2 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ ((reverseβ€˜(𝑆 ++ 𝑇))β€˜π‘₯) = (((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))β€˜π‘₯))
17217, 32, 171eqfnfvd 6990 1 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (reverseβ€˜(𝑆 ++ 𝑇)) = ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   βŠ† wss 3915   Fn wfn 6496  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  0cc0 11058  1c1 11059   + caddc 11061   βˆ’ cmin 11392  β„•0cn0 12420  β„€cz 12506  β„€β‰₯cuz 12770  ...cfz 13431  ..^cfzo 13574  β™―chash 14237  Word cword 14409   ++ cconcat 14465  reversecreverse 14653
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-concat 14466  df-reverse 14654
This theorem is referenced by:  gsumwrev  19154  efginvrel2  19516
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