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Theorem ccatswrd 14614
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 14591 . . . . . 6 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴)
21adantr 481 . . . . 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 14591 . . . . . 6 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴)
43adantr 481 . . . . 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 14520 . . . . 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 584 . . . 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 14474 . . . 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 14521 . . . . . . 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 584 . . . . . 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 483 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑆 ∈ Word 𝐴)
12 simpr1 1194 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ (0...π‘Œ))
13 simpr2 1195 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ (0...𝑍))
14 simpr3 1196 . . . . . . . . 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 13535 . . . . . . . . . . 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 495 . . . . . . . . 9 ((π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) β†’ π‘Œ ∈ (0...(β™―β€˜π‘†)))
1813, 14, 17syl2anc 584 . . . . . . . 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 14593 . . . . . . . 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 1371 . . . . . . 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 14593 . . . . . . . 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 1182 . . . . . . 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 7423 . . . . . 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 13498 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ β„€)
2524zcnd 12663 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ β„‚)
2612elfzelzd 13498 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ β„€)
2726zcnd 12663 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ β„‚)
2814elfzelzd 13498 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ β„€)
2928zcnd 12663 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ β„‚)
3025, 27, 29npncan3d 11603 . . . . . 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 2776 . . . . 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 7421 . . . 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 6640 . . 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 14591 . . . . 5 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) ∈ Word 𝐴)
3635adantr 481 . . . 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 14474 . . . 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 13535 . . . . . . . . 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 495 . . . . . . 7 ((𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍)) β†’ 𝑋 ∈ (0...𝑍))
4212, 13, 41syl2anc 584 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ (0...𝑍))
43 swrdlen 14593 . . . . . 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 1371 . . . . 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 7421 . . . 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 6640 . . 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 12667 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„€)
4948anim1ci 616 . . . . 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 13660 . . . . 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 724 . . . . . . 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 724 . . . . . . 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 7421 . . . . . . . . 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 2819 . . . . . . . 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 478 . . . . . . 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 14523 . . . . . . 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 1371 . . . . . 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 765 . . . . . . 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 1215 . . . . . . 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 481 . . . . . . 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 485 . . . . . . 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 14594 . . . . . . 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 1373 . . . . . 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 2772 . . . . 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 724 . . . . . . 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 724 . . . . . . 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 2772 . . . . . . . . . 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 7423 . . . . . . . . 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 2819 . . . . . . . 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 478 . . . . . . 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 14524 . . . . . . 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 1371 . . . . . 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 765 . . . . . . 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 1216 . . . . . . 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 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†’ 𝑍 ∈ (0...(β™―β€˜π‘†)))
7720oveq2d 7421 . . . . . . . . 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 481 . . . . . . . 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 7421 . . . . . . . . . . 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 2819 . . . . . . . . . 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 478 . . . . . . . . 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 12667 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑍 βˆ’ π‘Œ) ∈ β„€)
8382adantr 481 . . . . . . . . 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 13689 . . . . . . . . 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 584 . . . . . . . 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 2833 . . . . . . 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 14594 . . . . . . 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 1373 . . . . . 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 7420 . . . . . . . . 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 481 . . . . . . . 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 13628 . . . . . . . . . . 11 (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)) β†’ π‘₯ ∈ β„€)
9291zcnd 12663 . . . . . . . . . 10 (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)) β†’ π‘₯ ∈ β„‚)
9392adantl 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‘₯ ∈ β„‚)
9425, 27subcld 11567 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„‚)
9594adantr 481 . . . . . . . . 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 481 . . . . . . . . 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 11589 . . . . . . . 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 11572 . . . . . . . . . 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 7421 . . . . . . . . 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 481 . . . . . . . 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 2776 . . . . . . 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 6892 . . . . . 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 2776 . . . . 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 956 . . . 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 591 . . 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 765 . . . 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 481 . . . 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 1217 . . . 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 485 . . . 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 14594 . . . 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 1373 . . 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 2775 . 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 7032 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βŸ¨cop 4633   Fn wfn 6535  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  0cc0 11106   + caddc 11109   βˆ’ cmin 11440  β„€cz 12554  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460   ++ cconcat 14516   substr csubstr 14586
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-substr 14587
This theorem is referenced by:  swrds2  14887  efgredleme  19605
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