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Theorem ccatswrd 14120
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...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))

Proof of Theorem ccatswrd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 swrdcl 14097 . . . . . 6 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
21adantr 484 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
3 swrdcl 14097 . . . . . 6 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
43adantr 484 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5 ccatcl 14016 . . . . 5 (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
62, 4, 5syl2anc 587 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
7 wrdfn 13970 . . . 4 (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴 → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
86, 7syl 17 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
9 ccatlen 14017 . . . . . . 7 (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
102, 4, 9syl2anc 587 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
11 simpl 486 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
12 simpr1 1195 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑌))
13 simpr2 1196 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...𝑍))
14 simpr3 1197 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ (0...(♯‘𝑆)))
15 fzass4 13037 . . . . . . . . . . 11 ((𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
1615biimpri 231 . . . . . . . . . 10 ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))))
1716simpld 498 . . . . . . . . 9 ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ (0...(♯‘𝑆)))
1813, 14, 17syl2anc 587 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆)))
19 swrdlen 14099 . . . . . . . 8 ((𝑆 ∈ Word 𝐴𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌𝑋))
2011, 12, 18, 19syl3anc 1372 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌𝑋))
21 swrdlen 14099 . . . . . . . 8 ((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍𝑌))
22213adant3r1 1183 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍𝑌))
2320, 22oveq12d 7189 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = ((𝑌𝑋) + (𝑍𝑌)))
2413elfzelzd 13000 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
2524zcnd 12170 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ)
2612elfzelzd 13000 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℤ)
2726zcnd 12170 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℂ)
2814elfzelzd 13000 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℤ)
2928zcnd 12170 . . . . . . 7 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℂ)
3025, 27, 29npncan3d 11112 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌𝑋) + (𝑍𝑌)) = (𝑍𝑋))
3110, 23, 303eqtrd 2777 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍𝑋))
3231oveq2d 7187 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^(𝑍𝑋)))
3332fneq2d 6433 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(𝑍𝑋))))
348, 33mpbid 235 . 2 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(𝑍𝑋)))
35 swrdcl 14097 . . . . 5 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
3635adantr 484 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
37 wrdfn 13970 . . . 4 ((𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
3836, 37syl 17 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
39 fzass4 13037 . . . . . . . . 9 ((𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)))
4039biimpri 231 . . . . . . . 8 ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → (𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)))
4140simpld 498 . . . . . . 7 ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → 𝑋 ∈ (0...𝑍))
4212, 13, 41syl2anc 587 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑍))
43 swrdlen 14099 . . . . . 6 ((𝑆 ∈ Word 𝐴𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍𝑋))
4411, 42, 14, 43syl3anc 1372 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍𝑋))
4544oveq2d 7187 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))) = (0..^(𝑍𝑋)))
4645fneq2d 6433 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))) ↔ (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍𝑋))))
4738, 46mpbid 235 . 2 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍𝑋)))
4824, 26zsubcld 12174 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌𝑋) ∈ ℤ)
4948anim1ci 619 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → (𝑥 ∈ (0..^(𝑍𝑋)) ∧ (𝑌𝑋) ∈ ℤ))
50 fzospliti 13161 . . . . 5 ((𝑥 ∈ (0..^(𝑍𝑋)) ∧ (𝑌𝑋) ∈ ℤ) → (𝑥 ∈ (0..^(𝑌𝑋)) ∨ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))))
5149, 50syl 17 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → (𝑥 ∈ (0..^(𝑌𝑋)) ∨ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))))
521ad2antrr 726 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
533ad2antrr 726 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5420oveq2d 7187 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (0..^(𝑌𝑋)))
5554eleq2d 2818 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ↔ 𝑥 ∈ (0..^(𝑌𝑋))))
5655biimpar 481 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))))
57 ccatval1 14020 . . . . . . 7 (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥))
5852, 53, 56, 57syl3anc 1372 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥))
59 simpll 767 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑆 ∈ Word 𝐴)
60 simplr1 1216 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑋 ∈ (0...𝑌))
6118adantr 484 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑌 ∈ (0...(♯‘𝑆)))
62 simpr 488 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → 𝑥 ∈ (0..^(𝑌𝑋)))
63 swrdfv 14100 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
6459, 60, 61, 62, 63syl31anc 1374 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
6558, 64eqtrd 2773 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
661ad2antrr 726 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
673ad2antrr 726 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
6823, 30eqtrd 2773 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍𝑋))
6920, 68oveq12d 7189 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) = ((𝑌𝑋)..^(𝑍𝑋)))
7069eleq2d 2818 . . . . . . . 8 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))))
7170biimpar 481 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))))
72 ccatval2 14022 . . . . . . 7 (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))))
7366, 67, 71, 72syl3anc 1372 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))))
74 simpll 767 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑆 ∈ Word 𝐴)
75 simplr2 1217 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑌 ∈ (0...𝑍))
76 simplr3 1218 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑍 ∈ (0...(♯‘𝑆)))
7720oveq2d 7187 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌𝑋)))
7877adantr 484 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌𝑋)))
7930oveq2d 7187 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌𝑋)..^((𝑌𝑋) + (𝑍𝑌))) = ((𝑌𝑋)..^(𝑍𝑋)))
8079eleq2d 2818 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((𝑌𝑋)..^((𝑌𝑋) + (𝑍𝑌))) ↔ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))))
8180biimpar 481 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑥 ∈ ((𝑌𝑋)..^((𝑌𝑋) + (𝑍𝑌))))
8228, 24zsubcld 12174 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑍𝑌) ∈ ℤ)
8382adantr 484 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑍𝑌) ∈ ℤ)
84 fzosubel3 13190 . . . . . . . . 9 ((𝑥 ∈ ((𝑌𝑋)..^((𝑌𝑋) + (𝑍𝑌))) ∧ (𝑍𝑌) ∈ ℤ) → (𝑥 − (𝑌𝑋)) ∈ (0..^(𝑍𝑌)))
8581, 83, 84syl2anc 587 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑥 − (𝑌𝑋)) ∈ (0..^(𝑍𝑌)))
8678, 85eqeltrd 2833 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ∈ (0..^(𝑍𝑌)))
87 swrdfv 14100 . . . . . . 7 (((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ∈ (0..^(𝑍𝑌))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) = (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)))
8874, 75, 76, 86, 87syl31anc 1374 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) = (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)))
8977oveq1d 7186 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌𝑋)) + 𝑌))
9089adantr 484 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌𝑋)) + 𝑌))
91 elfzoelz 13130 . . . . . . . . . . 11 (𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋)) → 𝑥 ∈ ℤ)
9291zcnd 12170 . . . . . . . . . 10 (𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋)) → 𝑥 ∈ ℂ)
9392adantl 485 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑥 ∈ ℂ)
9425, 27subcld 11076 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌𝑋) ∈ ℂ)
9594adantr 484 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑌𝑋) ∈ ℂ)
9625adantr 484 . . . . . . . . 9 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → 𝑌 ∈ ℂ)
9793, 95, 96subadd23d 11098 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → ((𝑥 − (𝑌𝑋)) + 𝑌) = (𝑥 + (𝑌 − (𝑌𝑋))))
9825, 27nncand 11081 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − (𝑌𝑋)) = 𝑋)
9998oveq2d 7187 . . . . . . . . 9 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 + (𝑌 − (𝑌𝑋))) = (𝑥 + 𝑋))
10099adantr 484 . . . . . . . 8 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑥 + (𝑌 − (𝑌𝑋))) = (𝑥 + 𝑋))
10190, 97, 1003eqtrd 2777 . . . . . . 7 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = (𝑥 + 𝑋))
102101fveq2d 6679 . . . . . 6 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)) = (𝑆‘(𝑥 + 𝑋)))
10373, 88, 1023eqtrd 2777 . . . . 5 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
10465, 103jaodan 957 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ (𝑥 ∈ (0..^(𝑌𝑋)) ∨ 𝑥 ∈ ((𝑌𝑋)..^(𝑍𝑋)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
10551, 104syldan 594 . . 3 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
106 simpll 767 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → 𝑆 ∈ Word 𝐴)
10742adantr 484 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → 𝑋 ∈ (0...𝑍))
108 simplr3 1218 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → 𝑍 ∈ (0...(♯‘𝑆)))
109 simpr 488 . . . 4 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → 𝑥 ∈ (0..^(𝑍𝑋)))
110 swrdfv 14100 . . . 4 (((𝑆 ∈ Word 𝐴𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
111106, 107, 108, 109, 110syl31anc 1374 . . 3 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
112105, 111eqtr4d 2776 . 2 (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥))
11334, 47, 112eqfnfvd 6813 1 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 846  w3a 1088   = wceq 1542  wcel 2113  cop 4523   Fn wfn 6335  cfv 6340  (class class class)co 7171  cc 10614  0cc0 10616   + caddc 10619  cmin 10949  cz 12063  ...cfz 12982  ..^cfzo 13125  chash 13783  Word cword 13956   ++ cconcat 14012   substr csubstr 14092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-cnex 10672  ax-resscn 10673  ax-1cn 10674  ax-icn 10675  ax-addcl 10676  ax-addrcl 10677  ax-mulcl 10678  ax-mulrcl 10679  ax-mulcom 10680  ax-addass 10681  ax-mulass 10682  ax-distr 10683  ax-i2m1 10684  ax-1ne0 10685  ax-1rid 10686  ax-rnegex 10687  ax-rrecex 10688  ax-cnre 10689  ax-pre-lttri 10690  ax-pre-lttrn 10691  ax-pre-ltadd 10692  ax-pre-mulgt0 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7128  df-ov 7174  df-oprab 7175  df-mpo 7176  df-om 7601  df-1st 7715  df-2nd 7716  df-wrecs 7977  df-recs 8038  df-rdg 8076  df-1o 8132  df-er 8321  df-en 8557  df-dom 8558  df-sdom 8559  df-fin 8560  df-card 9442  df-pnf 10756  df-mnf 10757  df-xr 10758  df-ltxr 10759  df-le 10760  df-sub 10951  df-neg 10952  df-nn 11718  df-n0 11978  df-z 12064  df-uz 12326  df-fz 12983  df-fzo 13126  df-hash 13784  df-word 13957  df-concat 14013  df-substr 14093
This theorem is referenced by:  swrds2  14392  efgredleme  18988
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