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Theorem revccat 14712
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 14520 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (𝑆 ++ 𝑇) ∈ Word 𝐴)
2 revcl 14707 . . . 4 ((𝑆 ++ 𝑇) ∈ Word 𝐴 β†’ (reverseβ€˜(𝑆 ++ 𝑇)) ∈ Word 𝐴)
3 wrdfn 14474 . . . 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 14708 . . . . . . 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 14521 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(𝑆 ++ 𝑇)) = ((β™―β€˜π‘†) + (β™―β€˜π‘‡)))
8 lencl 14479 . . . . . . . 8 (𝑆 ∈ Word 𝐴 β†’ (β™―β€˜π‘†) ∈ β„•0)
98nn0cnd 12530 . . . . . . 7 (𝑆 ∈ Word 𝐴 β†’ (β™―β€˜π‘†) ∈ β„‚)
10 lencl 14479 . . . . . . . 8 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜π‘‡) ∈ β„•0)
1110nn0cnd 12530 . . . . . . 7 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜π‘‡) ∈ β„‚)
12 addcom 11396 . . . . . . 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 596 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
146, 7, 133eqtrd 2776 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇))) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
1514oveq2d 7421 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜(reverseβ€˜(𝑆 ++ 𝑇)))) = (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
1615fneq2d 6640 . . 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 14707 . . . . 5 (𝑇 ∈ Word 𝐴 β†’ (reverseβ€˜π‘‡) ∈ Word 𝐴)
19 revcl 14707 . . . . 5 (𝑆 ∈ Word 𝐴 β†’ (reverseβ€˜π‘†) ∈ Word 𝐴)
20 ccatcl 14520 . . . . 5 (((reverseβ€˜π‘‡) ∈ Word 𝐴 ∧ (reverseβ€˜π‘†) ∈ Word 𝐴) β†’ ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) ∈ Word 𝐴)
2118, 19, 20syl2anr 597 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)) ∈ Word 𝐴)
22 wrdfn 14474 . . . 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 14521 . . . . . . 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 597 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))) = ((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†))))
26 revlen 14708 . . . . . . 7 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜(reverseβ€˜π‘‡)) = (β™―β€˜π‘‡))
27 revlen 14708 . . . . . . 7 (𝑆 ∈ Word 𝐴 β†’ (β™―β€˜(reverseβ€˜π‘†)) = (β™―β€˜π‘†))
2826, 27oveqan12rd 7425 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(reverseβ€˜π‘‡)) + (β™―β€˜(reverseβ€˜π‘†))) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
2925, 28eqtrd 2772 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†))) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
3029oveq2d 7421 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜((reverseβ€˜π‘‡) ++ (reverseβ€˜π‘†)))) = (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
3130fneq2d 6640 . . 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 12580 . . . . 5 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜π‘‡) ∈ β„€)
3534adantl 482 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜π‘‡) ∈ β„€)
36 fzospliti 13660 . . . 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 597 . . 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 765 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ 𝑆 ∈ Word 𝐴)
39 simplr 767 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ 𝑇 ∈ Word 𝐴)
40 fzoval 13629 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜π‘‡)) = (0...((β™―β€˜π‘‡) βˆ’ 1)))
4342eleq2d 2819 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) ↔ π‘₯ ∈ (0...((β™―β€˜π‘‡) βˆ’ 1))))
4443biimpa 477 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ (0...((β™―β€˜π‘‡) βˆ’ 1)))
45 fznn0sub2 13604 . . . . . . . . 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 725 . . . . . . . 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 2836 . . . . . . 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 14525 . . . . . . 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 1371 . . . . . 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 2772 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(𝑆 ++ 𝑇)) = ((β™―β€˜π‘‡) + (β™―β€˜π‘†)))
5251oveq1d 7420 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) = (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1))
5311adantl 482 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜π‘‡) ∈ β„‚)
549adantr 481 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜π‘†) ∈ β„‚)
55 1cnd 11205 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ 1 ∈ β„‚)
5653, 54, 55addsubd 11588 . . . . . . . . . . 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 2772 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) = (((β™―β€˜π‘‡) βˆ’ 1) + (β™―β€˜π‘†)))
5857oveq1d 7420 . . . . . . . . 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 481 . . . . . . . 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 12601 . . . . . . . . . . . 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 12663 . . . . . . . . . 10 (𝑇 ∈ Word 𝐴 β†’ ((β™―β€˜π‘‡) βˆ’ 1) ∈ β„‚)
6362ad2antlr 725 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((β™―β€˜π‘‡) βˆ’ 1) ∈ β„‚)
649ad2antrr 724 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (β™―β€˜π‘†) ∈ β„‚)
65 elfzoelz 13628 . . . . . . . . . . 11 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ ∈ β„€)
6665zcnd 12663 . . . . . . . . . 10 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ ∈ β„‚)
6766adantl 482 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ β„‚)
6863, 64, 67addsubd 11588 . . . . . . . 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 2772 . . . . . . 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 6892 . . . . . 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 14709 . . . . . . 7 ((𝑇 ∈ Word 𝐴 ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((reverseβ€˜π‘‡)β€˜π‘₯) = (π‘‡β€˜(((β™―β€˜π‘‡) βˆ’ 1) βˆ’ π‘₯)))
7271adantll 712 . . . . . 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 2782 . . . . 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 12834 . . . . . . . . . 10 (𝑇 ∈ Word 𝐴 β†’ (β™―β€˜π‘‡) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)))
75 uzaddcl 12884 . . . . . . . . . 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 597 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)))
7751, 76eqeltrd 2833 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(𝑆 ++ 𝑇)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘‡)))
78 fzoss2 13656 . . . . . . . 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 3981 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ (0..^(β™―β€˜(𝑆 ++ 𝑇))))
81 revfv 14709 . . . . . 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 683 . . . . 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 725 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (reverseβ€˜π‘‡) ∈ Word 𝐴)
8419ad2antrr 724 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (reverseβ€˜π‘†) ∈ Word 𝐴)
8526adantl 482 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜(reverseβ€˜π‘‡)) = (β™―β€˜π‘‡))
8685oveq2d 7421 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜(reverseβ€˜π‘‡))) = (0..^(β™―β€˜π‘‡)))
8786eleq2d 2819 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (π‘₯ ∈ (0..^(β™―β€˜(reverseβ€˜π‘‡))) ↔ π‘₯ ∈ (0..^(β™―β€˜π‘‡))))
8887biimpar 478 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ (0..^(β™―β€˜(reverseβ€˜π‘‡))))
89 ccatval1 14523 . . . . . 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 1371 . . . . 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 2782 . . . 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 12580 . . . . . . . . . . . 12 (𝑆 ∈ Word 𝐴 β†’ (β™―β€˜π‘†) ∈ β„€)
93 peano2zm 12601 . . . . . . . . . . . 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 12663 . . . . . . . . . 10 (𝑆 ∈ Word 𝐴 β†’ ((β™―β€˜π‘†) βˆ’ 1) ∈ β„‚)
9695ad2antrr 724 . . . . . . . . 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 13628 . . . . . . . . . . 11 (π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) β†’ π‘₯ ∈ β„€)
9897zcnd 12663 . . . . . . . . . 10 (π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) β†’ π‘₯ ∈ β„‚)
9998adantl 482 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ π‘₯ ∈ β„‚)
10011ad2antlr 725 . . . . . . . . 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 11597 . . . . . . . 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 7421 . . . . . . . . . 10 (𝑇 ∈ Word 𝐴 β†’ (π‘₯ βˆ’ (β™―β€˜(reverseβ€˜π‘‡))) = (π‘₯ βˆ’ (β™―β€˜π‘‡)))
103102oveq2d 7421 . . . . . . . . 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 725 . . . . . . . 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 7420 . . . . . . . . . . 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 11588 . . . . . . . . . . 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 2772 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜(𝑆 ++ 𝑇)) βˆ’ 1) = (((β™―β€˜π‘†) βˆ’ 1) + (β™―β€˜π‘‡)))
108107oveq1d 7420 . . . . . . . . 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 481 . . . . . . . 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 2783 . . . . . . 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 6892 . . . . . 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 765 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ 𝑆 ∈ Word 𝐴)
113 simplr 767 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ 𝑇 ∈ Word 𝐴)
114 zaddcl 12598 . . . . . . . . . . 11 (((β™―β€˜π‘‡) ∈ β„€ ∧ (β™―β€˜π‘†) ∈ β„€) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„€)
11534, 92, 114syl2anr 597 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„€)
116 peano2zm 12601 . . . . . . . . . 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 13629 . . . . . . . . . . . 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 2819 . . . . . . . . . 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 477 . . . . . . . . 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 13562 . . . . . . . . 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 683 . . . . . . . 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 7420 . . . . . . . . 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 481 . . . . . . . 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 481 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (β™―β€˜π‘†) ∈ β„€)
127 fzoval 13629 . . . . . . . . . . 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 12663 . . . . . . . . . . . 12 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ 1) ∈ β„‚)
130129subidd 11555 . . . . . . . . . . 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 11188 . . . . . . . . . . . . . 14 (((β™―β€˜π‘‡) ∈ β„‚ ∧ (β™―β€˜π‘†) ∈ β„‚) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„‚)
13211, 9, 131syl2anr 597 . . . . . . . . . . . . 13 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡) + (β™―β€˜π‘†)) ∈ β„‚)
133132, 55, 53sub32d 11599 . . . . . . . . . . . 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 11463 . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . 13 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (((β™―β€˜π‘‡) + (β™―β€˜π‘†)) βˆ’ (β™―β€˜π‘‡)) = (β™―β€˜π‘†))
136135oveq1d 7420 . . . . . . . . . . . 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 2772 . . . . . . . . . . 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 7423 . . . . . . . . . 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 2775 . . . . . . . . 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 481 . . . . . . . 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 2848 . . . . . . 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 14523 . . . . . . 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 1371 . . . . . 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 483 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ 𝑆 ∈ Word 𝐴)
145102ad2antlr 725 . . . . . . . 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 13689 . . . . . . . . 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 597 . . . . . . . 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 2833 . . . . . . 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 14709 . . . . . . 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 683 . . . . . 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 2782 . . . . 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 13655 . . . . . . . . . . 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 12860 . . . . . . . . . . 11 β„•0 = (β„€β‰₯β€˜0)
155153, 154eleq2s 2851 . . . . . . . . . 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 482 . . . . . . . 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 7421 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ (0..^(β™―β€˜(𝑆 ++ 𝑇))) = (0..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))))
159157, 158sseqtrrd 4022 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) β†’ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†))) βŠ† (0..^(β™―β€˜(𝑆 ++ 𝑇))))
160159sselda 3981 . . . . . 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 683 . . . . 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 725 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (reverseβ€˜π‘‡) ∈ Word 𝐴)
16319ad2antrr 724 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ π‘₯ ∈ ((β™―β€˜π‘‡)..^((β™―β€˜π‘‡) + (β™―β€˜π‘†)))) β†’ (reverseβ€˜π‘†) ∈ Word 𝐴)
16485, 28oveq12d 7423 . . . . . . . 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 2819 . . . . . . 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 478 . . . . . 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 14524 . . . . . 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 1371 . . . . 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 2782 . . . 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 956 . . 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 591 . 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 7032 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   βŠ† wss 3947   Fn wfn 6535  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  0cc0 11106  1c1 11107   + caddc 11109   βˆ’ cmin 11440  β„•0cn0 12468  β„€cz 12554  β„€β‰₯cuz 12818  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460   ++ cconcat 14516  reversecreverse 14704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-concat 14517  df-reverse 14705
This theorem is referenced by:  gsumwrev  19227  efginvrel2  19589
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