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Theorem ccatswrd 14624
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 14601 . . . . . 6 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ∈ Word 𝐴)
21adantr 480 . . . . 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 14601 . . . . . 6 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©) ∈ Word 𝐴)
43adantr 480 . . . . 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 14530 . . . . 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 583 . . . 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 14484 . . . 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 14531 . . . . . . 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 583 . . . . . 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 482 . . . . . . . 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 13545 . . . . . . . . . . 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 494 . . . . . . . . 9 ((π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†))) β†’ π‘Œ ∈ (0...(β™―β€˜π‘†)))
1813, 14, 17syl2anc 583 . . . . . . . 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 14603 . . . . . . . 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 14603 . . . . . . . 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 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 13508 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ β„€)
2524zcnd 12671 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ π‘Œ ∈ β„‚)
2612elfzelzd 13508 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ β„€)
2726zcnd 12671 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ β„‚)
2814elfzelzd 13508 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ β„€)
2928zcnd 12671 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑍 ∈ β„‚)
3025, 27, 29npncan3d 11611 . . . . . 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 2770 . . . . 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 6637 . . 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 14601 . . . . 5 (𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘‹, π‘βŸ©) ∈ Word 𝐴)
3635adantr 480 . . . 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 14484 . . . 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 13545 . . . . . . . . 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 494 . . . . . . 7 ((𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍)) β†’ 𝑋 ∈ (0...𝑍))
4212, 13, 41syl2anc 583 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ 𝑋 ∈ (0...𝑍))
43 swrdlen 14603 . . . . . 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 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 6637 . . 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 12675 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„€)
4948anim1ci 615 . . . . 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 13670 . . . . 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 723 . . . . . . 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 723 . . . . . . 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 2813 . . . . . . . 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 477 . . . . . . 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 14533 . . . . . . 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 764 . . . . . . 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 480 . . . . . . 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 484 . . . . . . 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 14604 . . . . . . 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 2766 . . . . 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 723 . . . . . . 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 723 . . . . . . 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 2766 . . . . . . . . . 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 2813 . . . . . . . 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 477 . . . . . . 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 14534 . . . . . . 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 764 . . . . . . 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 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 480 . . . . . . . 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 2813 . . . . . . . . . 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 477 . . . . . . . . 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 12675 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (𝑍 βˆ’ π‘Œ) ∈ β„€)
8382adantr 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†’ (𝑍 βˆ’ π‘Œ) ∈ β„€)
84 fzosubel3 13699 . . . . . . . . 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 583 . . . . . . . 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 2827 . . . . . . 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 14604 . . . . . . 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 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 480 . . . . . . . 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 13638 . . . . . . . . . . 11 (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)) β†’ π‘₯ ∈ β„€)
9291zcnd 12671 . . . . . . . . . 10 (π‘₯ ∈ ((π‘Œ βˆ’ 𝑋)..^(𝑍 βˆ’ 𝑋)) β†’ π‘₯ ∈ β„‚)
9392adantl 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‘₯ ∈ β„‚)
9425, 27subcld 11575 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ (π‘Œ βˆ’ 𝑋) ∈ β„‚)
9594adantr 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‘Œ βˆ’ 𝑋) ∈ β„‚)
9625adantr 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‘Œ ∈ β„‚)
9793, 95, 96subadd23d 11597 . . . . . . . 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 11580 . . . . . . . . . 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 480 . . . . . . . 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 2770 . . . . . . 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 6889 . . . . . 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 2770 . . . . 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 954 . . . 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 590 . . 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 764 . . . 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 480 . . . 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 484 . . . 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 14604 . . . 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 2769 . 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 7029 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 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4629   Fn wfn 6532  β€˜cfv 6537  (class class class)co 7405  β„‚cc 11110  0cc0 11112   + caddc 11115   βˆ’ cmin 11448  β„€cz 12562  ...cfz 13490  ..^cfzo 13633  β™―chash 14295  Word cword 14470   ++ cconcat 14526   substr csubstr 14596
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-concat 14527  df-substr 14597
This theorem is referenced by:  swrds2  14897  efgredleme  19663
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