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Theorem ccatswrd 14660
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 14637 . . . . . 6 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴)
21adantr 479 . . . . 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 14637 . . . . . 6 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴)
43adantr 479 . . . . 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 14566 . . . . 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 582 . . . 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 14520 . . . 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 14567 . . . . . . 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 582 . . . . . 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 481 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑆 ∈ Word 𝐴)
12 simpr1 1191 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ (0...π‘Œ))
13 simpr2 1192 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ (0...𝑍))
14 simpr3 1193 . . . . . . . . 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 13581 . . . . . . . . . . 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 493 . . . . . . . . 9 ((π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) β†’ π‘Œ ∈ (0...(β™―β€˜π‘†)))
1813, 14, 17syl2anc 582 . . . . . . . 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 14639 . . . . . . . 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 1368 . . . . . . 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 14639 . . . . . . . 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 1179 . . . . . . 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 7444 . . . . . 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 13544 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ β„€)
2524zcnd 12707 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ β„‚)
2612elfzelzd 13544 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ β„€)
2726zcnd 12707 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ β„‚)
2814elfzelzd 13544 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ β„€)
2928zcnd 12707 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ β„‚)
3025, 27, 29npncan3d 11647 . . . . . 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 2772 . . . . 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 7442 . . . 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 6653 . . 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 14637 . . . . 5 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) ∈ Word 𝐴)
3635adantr 479 . . . 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 14520 . . . 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 13581 . . . . . . . . 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 493 . . . . . . 7 ((𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍)) β†’ 𝑋 ∈ (0...𝑍))
4212, 13, 41syl2anc 582 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ (0...𝑍))
43 swrdlen 14639 . . . . . 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 1368 . . . . 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 7442 . . . 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 6653 . . 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 12711 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„€)
4948anim1ci 614 . . . . 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 13706 . . . . 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 7442 . . . . . . . . 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 2815 . . . . . . . 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 476 . . . . . . 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 14569 . . . . . . 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 1368 . . . . . 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 1212 . . . . . . 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 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‘Œ ∈ (0...(β™―β€˜π‘†)))
62 simpr 483 . . . . . . 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 14640 . . . . . . 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 1370 . . . . . 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 2768 . . . . 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 2768 . . . . . . . . . 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 7444 . . . . . . . . 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 2815 . . . . . . . 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 476 . . . . . . 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 14570 . . . . . . 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 1368 . . . . . 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 1213 . . . . . . 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 1214 . . . . . . 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 7442 . . . . . . . . 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 479 . . . . . . . 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 7442 . . . . . . . . . . 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 2815 . . . . . . . . . 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 476 . . . . . . . . 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 12711 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑍 βˆ’ π‘Œ) ∈ β„€)
8382adantr 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†’ (𝑍 βˆ’ π‘Œ) ∈ β„€)
84 fzosubel3 13735 . . . . . . . . 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 582 . . . . . . . 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 2829 . . . . . . 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 14640 . . . . . . 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 1370 . . . . . 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 7441 . . . . . . . . 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 479 . . . . . . . 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 13674 . . . . . . . . . . 11 (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)) β†’ π‘₯ ∈ β„€)
9291zcnd 12707 . . . . . . . . . 10 (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)) β†’ π‘₯ ∈ β„‚)
9392adantl 480 . . . . . . . . 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 11611 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„‚)
9594adantr 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‘Œ βˆ’ 𝑋) ∈ β„‚)
9625adantr 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‘Œ ∈ β„‚)
9793, 95, 96subadd23d 11633 . . . . . . . 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 11616 . . . . . . . . . 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 7442 . . . . . . . . 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 479 . . . . . . . 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 2772 . . . . . . 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 6906 . . . . . 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 2772 . . . . 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 955 . . . 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 589 . . 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 479 . . . 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 1214 . . . 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 483 . . . 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 14640 . . . 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 1370 . . 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 2771 . 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 7048 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 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4638   Fn wfn 6548  β€˜cfv 6553  (class class class)co 7426  β„‚cc 11146  0cc0 11148   + caddc 11151   βˆ’ cmin 11484  β„€cz 12598  ...cfz 13526  ..^cfzo 13669  β™―chash 14331  Word cword 14506   ++ cconcat 14562   substr csubstr 14632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-card 9972  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-n0 12513  df-z 12599  df-uz 12863  df-fz 13527  df-fzo 13670  df-hash 14332  df-word 14507  df-concat 14563  df-substr 14633
This theorem is referenced by:  swrds2  14933  efgredleme  19712
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