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Theorem ccatswrd 14563
Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Assertion
Ref Expression
ccatswrd ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) = (𝑆 substr βŸ¨π‘‹, π‘βŸ©))

Proof of Theorem ccatswrd
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 swrdcl 14540 . . . . . 6 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴)
21adantr 482 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴)
3 swrdcl 14540 . . . . . 6 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴)
43adantr 482 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴)
5 ccatcl 14469 . . . . 5 (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴 ∧ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) ∈ Word 𝐴)
62, 4, 5syl2anc 585 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) ∈ Word 𝐴)
7 wrdfn 14423 . . . 4 (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) ∈ Word 𝐴 β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) Fn (0..^(β™―β€˜((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)))))
86, 7syl 17 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) Fn (0..^(β™―β€˜((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)))))
9 ccatlen 14470 . . . . . . 7 (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴 ∧ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴) β†’ (β™―β€˜((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))) = ((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) + (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©))))
102, 4, 9syl2anc 585 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (β™―β€˜((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))) = ((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) + (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©))))
11 simpl 484 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑆 ∈ Word 𝐴)
12 simpr1 1195 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ (0...π‘Œ))
13 simpr2 1196 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ (0...𝑍))
14 simpr3 1197 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ (0...(β™―β€˜π‘†)))
15 fzass4 13486 . . . . . . . . . . 11 ((π‘Œ ∈ (0...(β™―β€˜π‘†)) ∧ 𝑍 ∈ (π‘Œ...(β™―β€˜π‘†))) ↔ (π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))))
1615biimpri 227 . . . . . . . . . 10 ((π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) β†’ (π‘Œ ∈ (0...(β™―β€˜π‘†)) ∧ 𝑍 ∈ (π‘Œ...(β™―β€˜π‘†))))
1716simpld 496 . . . . . . . . 9 ((π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) β†’ π‘Œ ∈ (0...(β™―β€˜π‘†)))
1813, 14, 17syl2anc 585 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ (0...(β™―β€˜π‘†)))
19 swrdlen 14542 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...(β™―β€˜π‘†))) β†’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) = (π‘Œ βˆ’ 𝑋))
2011, 12, 18, 19syl3anc 1372 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) = (π‘Œ βˆ’ 𝑋))
21 swrdlen 14542 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) β†’ (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) = (𝑍 βˆ’ π‘Œ))
22213adant3r1 1183 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) = (𝑍 βˆ’ π‘Œ))
2320, 22oveq12d 7380 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) + (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©))) = ((π‘Œ βˆ’ 𝑋) + (𝑍 βˆ’ π‘Œ)))
2413elfzelzd 13449 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ β„€)
2524zcnd 12615 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ β„‚)
2612elfzelzd 13449 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ β„€)
2726zcnd 12615 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ β„‚)
2814elfzelzd 13449 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ β„€)
2928zcnd 12615 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ β„‚)
3025, 27, 29npncan3d 11555 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((π‘Œ βˆ’ 𝑋) + (𝑍 βˆ’ π‘Œ)) = (𝑍 βˆ’ 𝑋))
3110, 23, 303eqtrd 2781 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (β™―β€˜((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))) = (𝑍 βˆ’ 𝑋))
3231oveq2d 7378 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (0..^(β™―β€˜((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)))) = (0..^(𝑍 βˆ’ 𝑋)))
3332fneq2d 6601 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) Fn (0..^(β™―β€˜((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)))) ↔ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) Fn (0..^(𝑍 βˆ’ 𝑋))))
348, 33mpbid 231 . 2 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) Fn (0..^(𝑍 βˆ’ 𝑋)))
35 swrdcl 14540 . . . . 5 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) ∈ Word 𝐴)
3635adantr 482 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) ∈ Word 𝐴)
37 wrdfn 14423 . . . 4 ((𝑆 substr βŸ¨π‘‹, π‘βŸ©) ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) Fn (0..^(β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘βŸ©))))
3836, 37syl 17 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) Fn (0..^(β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘βŸ©))))
39 fzass4 13486 . . . . . . . . 9 ((𝑋 ∈ (0...𝑍) ∧ π‘Œ ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍)))
4039biimpri 227 . . . . . . . 8 ((𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍)) β†’ (𝑋 ∈ (0...𝑍) ∧ π‘Œ ∈ (𝑋...𝑍)))
4140simpld 496 . . . . . . 7 ((𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍)) β†’ 𝑋 ∈ (0...𝑍))
4212, 13, 41syl2anc 585 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ (0...𝑍))
43 swrdlen 14542 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) β†’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘βŸ©)) = (𝑍 βˆ’ 𝑋))
4411, 42, 14, 43syl3anc 1372 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘βŸ©)) = (𝑍 βˆ’ 𝑋))
4544oveq2d 7378 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (0..^(β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘βŸ©))) = (0..^(𝑍 βˆ’ 𝑋)))
4645fneq2d 6601 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘βŸ©) Fn (0..^(β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘βŸ©))) ↔ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) Fn (0..^(𝑍 βˆ’ 𝑋))))
4738, 46mpbid 231 . 2 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) Fn (0..^(𝑍 βˆ’ 𝑋)))
4824, 26zsubcld 12619 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„€)
4948anim1ci 617 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ (π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋)) ∧ (π‘Œ βˆ’ 𝑋) ∈ β„€))
50 fzospliti 13611 . . . . 5 ((π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋)) ∧ (π‘Œ βˆ’ 𝑋) ∈ β„€) β†’ (π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋)) ∨ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))))
5149, 50syl 17 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ (π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋)) ∨ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))))
521ad2antrr 725 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ (𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴)
533ad2antrr 725 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴)
5420oveq2d 7378 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (0..^(β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) = (0..^(π‘Œ βˆ’ 𝑋)))
5554eleq2d 2824 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘₯ ∈ (0..^(β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) ↔ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))))
5655biimpar 479 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ π‘₯ ∈ (0..^(β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))))
57 ccatval1 14472 . . . . . . 7 (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴 ∧ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴 ∧ π‘₯ ∈ (0..^(β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)β€˜π‘₯))
5852, 53, 56, 57syl3anc 1372 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)β€˜π‘₯))
59 simpll 766 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ 𝑆 ∈ Word 𝐴)
60 simplr1 1216 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ 𝑋 ∈ (0...π‘Œ))
6118adantr 482 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ π‘Œ ∈ (0...(β™―β€˜π‘†)))
62 simpr 486 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋)))
63 swrdfv 14543 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...(β™―β€˜π‘†))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)β€˜π‘₯) = (π‘†β€˜(π‘₯ + 𝑋)))
6459, 60, 61, 62, 63syl31anc 1374 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)β€˜π‘₯) = (π‘†β€˜(π‘₯ + 𝑋)))
6558, 64eqtrd 2777 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = (π‘†β€˜(π‘₯ + 𝑋)))
661ad2antrr 725 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴)
673ad2antrr 725 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴)
6823, 30eqtrd 2777 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) + (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©))) = (𝑍 βˆ’ 𝑋))
6920, 68oveq12d 7380 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))..^((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) + (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©)))) = ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)))
7069eleq2d 2824 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘₯ ∈ ((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))..^((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) + (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©)))) ↔ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))))
7170biimpar 479 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ π‘₯ ∈ ((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))..^((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) + (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©)))))
72 ccatval2 14473 . . . . . . 7 (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴 ∧ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴 ∧ π‘₯ ∈ ((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))..^((β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)) + (β™―β€˜(𝑆 substr βŸ¨π‘Œ, π‘βŸ©))))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = ((𝑆 substr βŸ¨π‘Œ, π‘βŸ©)β€˜(π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)))))
7366, 67, 71, 72syl3anc 1372 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = ((𝑆 substr βŸ¨π‘Œ, π‘βŸ©)β€˜(π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)))))
74 simpll 766 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ 𝑆 ∈ Word 𝐴)
75 simplr2 1217 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ π‘Œ ∈ (0...𝑍))
76 simplr3 1218 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ 𝑍 ∈ (0...(β™―β€˜π‘†)))
7720oveq2d 7378 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) = (π‘₯ βˆ’ (π‘Œ βˆ’ 𝑋)))
7877adantr 482 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) = (π‘₯ βˆ’ (π‘Œ βˆ’ 𝑋)))
7930oveq2d 7378 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((π‘Œ βˆ’ 𝑋)..^((π‘Œ βˆ’ 𝑋) + (𝑍 βˆ’ π‘Œ))) = ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)))
8079eleq2d 2824 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^((π‘Œ βˆ’ 𝑋) + (𝑍 βˆ’ π‘Œ))) ↔ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))))
8180biimpar 479 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^((π‘Œ βˆ’ 𝑋) + (𝑍 βˆ’ π‘Œ))))
8228, 24zsubcld 12619 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑍 βˆ’ π‘Œ) ∈ β„€)
8382adantr 482 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (𝑍 βˆ’ π‘Œ) ∈ β„€)
84 fzosubel3 13640 . . . . . . . . 9 ((π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^((π‘Œ βˆ’ 𝑋) + (𝑍 βˆ’ π‘Œ))) ∧ (𝑍 βˆ’ π‘Œ) ∈ β„€) β†’ (π‘₯ βˆ’ (π‘Œ βˆ’ 𝑋)) ∈ (0..^(𝑍 βˆ’ π‘Œ)))
8581, 83, 84syl2anc 585 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (π‘₯ βˆ’ (π‘Œ βˆ’ 𝑋)) ∈ (0..^(𝑍 βˆ’ π‘Œ)))
8678, 85eqeltrd 2838 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) ∈ (0..^(𝑍 βˆ’ π‘Œ)))
87 swrdfv 14543 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) ∧ (π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) ∈ (0..^(𝑍 βˆ’ π‘Œ))) β†’ ((𝑆 substr βŸ¨π‘Œ, π‘βŸ©)β€˜(π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)))) = (π‘†β€˜((π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) + π‘Œ)))
8874, 75, 76, 86, 87syl31anc 1374 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ ((𝑆 substr βŸ¨π‘Œ, π‘βŸ©)β€˜(π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©)))) = (π‘†β€˜((π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) + π‘Œ)))
8977oveq1d 7377 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) + π‘Œ) = ((π‘₯ βˆ’ (π‘Œ βˆ’ 𝑋)) + π‘Œ))
9089adantr 482 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ ((π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) + π‘Œ) = ((π‘₯ βˆ’ (π‘Œ βˆ’ 𝑋)) + π‘Œ))
91 elfzoelz 13579 . . . . . . . . . . 11 (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)) β†’ π‘₯ ∈ β„€)
9291zcnd 12615 . . . . . . . . . 10 (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)) β†’ π‘₯ ∈ β„‚)
9392adantl 483 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ π‘₯ ∈ β„‚)
9425, 27subcld 11519 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„‚)
9594adantr 482 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„‚)
9625adantr 482 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ π‘Œ ∈ β„‚)
9793, 95, 96subadd23d 11541 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ ((π‘₯ βˆ’ (π‘Œ βˆ’ 𝑋)) + π‘Œ) = (π‘₯ + (π‘Œ βˆ’ (π‘Œ βˆ’ 𝑋))))
9825, 27nncand 11524 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ (π‘Œ βˆ’ 𝑋)) = 𝑋)
9998oveq2d 7378 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘₯ + (π‘Œ βˆ’ (π‘Œ βˆ’ 𝑋))) = (π‘₯ + 𝑋))
10099adantr 482 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (π‘₯ + (π‘Œ βˆ’ (π‘Œ βˆ’ 𝑋))) = (π‘₯ + 𝑋))
10190, 97, 1003eqtrd 2781 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ ((π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) + π‘Œ) = (π‘₯ + 𝑋))
102101fveq2d 6851 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (π‘†β€˜((π‘₯ βˆ’ (β™―β€˜(𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©))) + π‘Œ)) = (π‘†β€˜(π‘₯ + 𝑋)))
10373, 88, 1023eqtrd 2781 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = (π‘†β€˜(π‘₯ + 𝑋)))
10465, 103jaodan 957 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ (π‘₯ ∈ (0..^(π‘Œ βˆ’ 𝑋)) ∨ π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = (π‘†β€˜(π‘₯ + 𝑋)))
10551, 104syldan 592 . . 3 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = (π‘†β€˜(π‘₯ + 𝑋)))
106 simpll 766 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ 𝑆 ∈ Word 𝐴)
10742adantr 482 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ 𝑋 ∈ (0...𝑍))
108 simplr3 1218 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ 𝑍 ∈ (0...(β™―β€˜π‘†)))
109 simpr 486 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋)))
110 swrdfv 14543 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘βŸ©)β€˜π‘₯) = (π‘†β€˜(π‘₯ + 𝑋)))
111106, 107, 108, 109, 110syl31anc 1374 . . 3 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘βŸ©)β€˜π‘₯) = (π‘†β€˜(π‘₯ + 𝑋)))
112105, 111eqtr4d 2780 . 2 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) ∧ π‘₯ ∈ (0..^(𝑍 βˆ’ 𝑋))) β†’ (((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©))β€˜π‘₯) = ((𝑆 substr βŸ¨π‘‹, π‘βŸ©)β€˜π‘₯))
11334, 47, 112eqfnfvd 6990 1 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) = (𝑆 substr βŸ¨π‘‹, π‘βŸ©))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4597   Fn wfn 6496  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  0cc0 11058   + caddc 11061   βˆ’ cmin 11392  β„€cz 12506  ...cfz 13431  ..^cfzo 13574  β™―chash 14237  Word cword 14409   ++ cconcat 14465   substr csubstr 14535
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-substr 14536
This theorem is referenced by:  swrds2  14836  efgredleme  19532
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