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Theorem ccatrn 14539
Description: The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
ccatrn ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ran (𝑆 ++ 𝑇) = (ran 𝑆 βˆͺ ran 𝑇))
Proof of Theorem ccatrn
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ccatvalfn 14531 . . . 4 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (𝑆 ++ 𝑇) Fn (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))))
2 lencl 14483 . . . . . . . . . . . 12 (𝑆 ∈ Word 𝐡 β†’ (β™―β€˜π‘†) ∈ β„•0)
3 nn0uz 12864 . . . . . . . . . . . 12 β„•0 = (β„€β‰₯β€˜0)
42, 3eleqtrdi 2844 . . . . . . . . . . 11 (𝑆 ∈ Word 𝐡 β†’ (β™―β€˜π‘†) ∈ (β„€β‰₯β€˜0))
54adantr 482 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (β™―β€˜π‘†) ∈ (β„€β‰₯β€˜0))
62nn0zd 12584 . . . . . . . . . . . 12 (𝑆 ∈ Word 𝐡 β†’ (β™―β€˜π‘†) ∈ β„€)
76uzidd 12838 . . . . . . . . . . 11 (𝑆 ∈ Word 𝐡 β†’ (β™―β€˜π‘†) ∈ (β„€β‰₯β€˜(β™―β€˜π‘†)))
8 lencl 14483 . . . . . . . . . . 11 (𝑇 ∈ Word 𝐡 β†’ (β™―β€˜π‘‡) ∈ β„•0)
9 uzaddcl 12888 . . . . . . . . . . 11 (((β™―β€˜π‘†) ∈ (β„€β‰₯β€˜(β™―β€˜π‘†)) ∧ (β™―β€˜π‘‡) ∈ β„•0) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘†)))
107, 8, 9syl2an 597 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘†)))
11 elfzuzb 13495 . . . . . . . . . 10 ((β™―β€˜π‘†) ∈ (0...((β™―β€˜π‘†) + (β™―β€˜π‘‡))) ↔ ((β™―β€˜π‘†) ∈ (β„€β‰₯β€˜0) ∧ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘†))))
125, 10, 11sylanbrc 584 . . . . . . . . 9 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (β™―β€˜π‘†) ∈ (0...((β™―β€˜π‘†) + (β™―β€˜π‘‡))))
13 fzosplit 13665 . . . . . . . . 9 ((β™―β€˜π‘†) ∈ (0...((β™―β€˜π‘†) + (β™―β€˜π‘‡))) β†’ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) = ((0..^(β™―β€˜π‘†)) βˆͺ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))))
1412, 13syl 17 . . . . . . . 8 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) = ((0..^(β™―β€˜π‘†)) βˆͺ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))))
1514eleq2d 2820 . . . . . . 7 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) ↔ π‘₯ ∈ ((0..^(β™―β€˜π‘†)) βˆͺ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))))))
16 elun 4149 . . . . . . 7 (π‘₯ ∈ ((0..^(β™―β€˜π‘†)) βˆͺ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) ↔ (π‘₯ ∈ (0..^(β™―β€˜π‘†)) ∨ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))))
1715, 16bitrdi 287 . . . . . 6 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) ↔ (π‘₯ ∈ (0..^(β™―β€˜π‘†)) ∨ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))))))
18 ccatval1 14527 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡 ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) = (π‘†β€˜π‘₯))
19183expa 1119 . . . . . . . . 9 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) = (π‘†β€˜π‘₯))
20 ssun1 4173 . . . . . . . . . 10 ran 𝑆 βŠ† (ran 𝑆 βˆͺ ran 𝑇)
21 wrdfn 14478 . . . . . . . . . . . 12 (𝑆 ∈ Word 𝐡 β†’ 𝑆 Fn (0..^(β™―β€˜π‘†)))
2221adantr 482 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ 𝑆 Fn (0..^(β™―β€˜π‘†)))
23 fnfvelrn 7083 . . . . . . . . . . 11 ((𝑆 Fn (0..^(β™―β€˜π‘†)) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ (π‘†β€˜π‘₯) ∈ ran 𝑆)
2422, 23sylan 581 . . . . . . . . . 10 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ (π‘†β€˜π‘₯) ∈ ran 𝑆)
2520, 24sselid 3981 . . . . . . . . 9 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ (π‘†β€˜π‘₯) ∈ (ran 𝑆 βˆͺ ran 𝑇))
2619, 25eqeltrd 2834 . . . . . . . 8 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) ∈ (ran 𝑆 βˆͺ ran 𝑇))
27 ccatval2 14528 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡 ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) = (π‘‡β€˜(π‘₯ βˆ’ (β™―β€˜π‘†))))
28273expa 1119 . . . . . . . . 9 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) = (π‘‡β€˜(π‘₯ βˆ’ (β™―β€˜π‘†))))
29 ssun2 4174 . . . . . . . . . 10 ran 𝑇 βŠ† (ran 𝑆 βˆͺ ran 𝑇)
30 wrdfn 14478 . . . . . . . . . . . 12 (𝑇 ∈ Word 𝐡 β†’ 𝑇 Fn (0..^(β™―β€˜π‘‡)))
3130adantl 483 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ 𝑇 Fn (0..^(β™―β€˜π‘‡)))
32 elfzouz 13636 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) β†’ π‘₯ ∈ (β„€β‰₯β€˜(β™―β€˜π‘†)))
33 uznn0sub 12861 . . . . . . . . . . . . . . 15 (π‘₯ ∈ (β„€β‰₯β€˜(β™―β€˜π‘†)) β†’ (π‘₯ βˆ’ (β™―β€˜π‘†)) ∈ β„•0)
3432, 33syl 17 . . . . . . . . . . . . . 14 (π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) β†’ (π‘₯ βˆ’ (β™―β€˜π‘†)) ∈ β„•0)
3534, 3eleqtrdi 2844 . . . . . . . . . . . . 13 (π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) β†’ (π‘₯ βˆ’ (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜0))
3635adantl 483 . . . . . . . . . . . 12 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ (π‘₯ βˆ’ (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜0))
378nn0zd 12584 . . . . . . . . . . . . 13 (𝑇 ∈ Word 𝐡 β†’ (β™―β€˜π‘‡) ∈ β„€)
3837ad2antlr 726 . . . . . . . . . . . 12 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ (β™―β€˜π‘‡) ∈ β„€)
39 elfzolt2 13641 . . . . . . . . . . . . . 14 (π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) β†’ π‘₯ < ((β™―β€˜π‘†) + (β™―β€˜π‘‡)))
4039adantl 483 . . . . . . . . . . . . 13 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ π‘₯ < ((β™―β€˜π‘†) + (β™―β€˜π‘‡)))
41 elfzoelz 13632 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) β†’ π‘₯ ∈ β„€)
4241zred 12666 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) β†’ π‘₯ ∈ ℝ)
4342adantl 483 . . . . . . . . . . . . . 14 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ π‘₯ ∈ ℝ)
442nn0red 12533 . . . . . . . . . . . . . . 15 (𝑆 ∈ Word 𝐡 β†’ (β™―β€˜π‘†) ∈ ℝ)
4544ad2antrr 725 . . . . . . . . . . . . . 14 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ (β™―β€˜π‘†) ∈ ℝ)
468nn0red 12533 . . . . . . . . . . . . . . 15 (𝑇 ∈ Word 𝐡 β†’ (β™―β€˜π‘‡) ∈ ℝ)
4746ad2antlr 726 . . . . . . . . . . . . . 14 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ (β™―β€˜π‘‡) ∈ ℝ)
4843, 45, 47ltsubadd2d 11812 . . . . . . . . . . . . 13 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ ((π‘₯ βˆ’ (β™―β€˜π‘†)) < (β™―β€˜π‘‡) ↔ π‘₯ < ((β™―β€˜π‘†) + (β™―β€˜π‘‡))))
4940, 48mpbird 257 . . . . . . . . . . . 12 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ (π‘₯ βˆ’ (β™―β€˜π‘†)) < (β™―β€˜π‘‡))
50 elfzo2 13635 . . . . . . . . . . . 12 ((π‘₯ βˆ’ (β™―β€˜π‘†)) ∈ (0..^(β™―β€˜π‘‡)) ↔ ((π‘₯ βˆ’ (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜0) ∧ (β™―β€˜π‘‡) ∈ β„€ ∧ (π‘₯ βˆ’ (β™―β€˜π‘†)) < (β™―β€˜π‘‡)))
5136, 38, 49, 50syl3anbrc 1344 . . . . . . . . . . 11 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ (π‘₯ βˆ’ (β™―β€˜π‘†)) ∈ (0..^(β™―β€˜π‘‡)))
52 fnfvelrn 7083 . . . . . . . . . . 11 ((𝑇 Fn (0..^(β™―β€˜π‘‡)) ∧ (π‘₯ βˆ’ (β™―β€˜π‘†)) ∈ (0..^(β™―β€˜π‘‡))) β†’ (π‘‡β€˜(π‘₯ βˆ’ (β™―β€˜π‘†))) ∈ ran 𝑇)
5331, 51, 52syl2an2r 684 . . . . . . . . . 10 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ (π‘‡β€˜(π‘₯ βˆ’ (β™―β€˜π‘†))) ∈ ran 𝑇)
5429, 53sselid 3981 . . . . . . . . 9 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ (π‘‡β€˜(π‘₯ βˆ’ (β™―β€˜π‘†))) ∈ (ran 𝑆 βˆͺ ran 𝑇))
5528, 54eqeltrd 2834 . . . . . . . 8 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) ∈ (ran 𝑆 βˆͺ ran 𝑇))
5626, 55jaodan 957 . . . . . . 7 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ (π‘₯ ∈ (0..^(β™―β€˜π‘†)) ∨ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) ∈ (ran 𝑆 βˆͺ ran 𝑇))
5756ex 414 . . . . . 6 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ((π‘₯ ∈ (0..^(β™―β€˜π‘†)) ∨ π‘₯ ∈ ((β™―β€˜π‘†)..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) ∈ (ran 𝑆 βˆͺ ran 𝑇)))
5817, 57sylbid 239 . . . . 5 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) ∈ (ran 𝑆 βˆͺ ran 𝑇)))
5958ralrimiv 3146 . . . 4 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ βˆ€π‘₯ ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))((𝑆 ++ 𝑇)β€˜π‘₯) ∈ (ran 𝑆 βˆͺ ran 𝑇))
60 ffnfv 7118 . . . 4 ((𝑆 ++ 𝑇):(0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))⟢(ran 𝑆 βˆͺ ran 𝑇) ↔ ((𝑆 ++ 𝑇) Fn (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) ∧ βˆ€π‘₯ ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))((𝑆 ++ 𝑇)β€˜π‘₯) ∈ (ran 𝑆 βˆͺ ran 𝑇)))
611, 59, 60sylanbrc 584 . . 3 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (𝑆 ++ 𝑇):(0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))⟢(ran 𝑆 βˆͺ ran 𝑇))
6261frnd 6726 . 2 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ran (𝑆 ++ 𝑇) βŠ† (ran 𝑆 βˆͺ ran 𝑇))
63 fzoss2 13660 . . . . . . . . . 10 (((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ (β„€β‰₯β€˜(β™―β€˜π‘†)) β†’ (0..^(β™―β€˜π‘†)) βŠ† (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))))
6410, 63syl 17 . . . . . . . . 9 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (0..^(β™―β€˜π‘†)) βŠ† (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))))
6564sselda 3983 . . . . . . . 8 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ π‘₯ ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))))
66 fnfvelrn 7083 . . . . . . . 8 (((𝑆 ++ 𝑇) Fn (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) ∈ ran (𝑆 ++ 𝑇))
671, 65, 66syl2an2r 684 . . . . . . 7 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ ((𝑆 ++ 𝑇)β€˜π‘₯) ∈ ran (𝑆 ++ 𝑇))
6819, 67eqeltrrd 2835 . . . . . 6 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘†))) β†’ (π‘†β€˜π‘₯) ∈ ran (𝑆 ++ 𝑇))
6968ralrimiva 3147 . . . . 5 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ βˆ€π‘₯ ∈ (0..^(β™―β€˜π‘†))(π‘†β€˜π‘₯) ∈ ran (𝑆 ++ 𝑇))
70 ffnfv 7118 . . . . 5 (𝑆:(0..^(β™―β€˜π‘†))⟢ran (𝑆 ++ 𝑇) ↔ (𝑆 Fn (0..^(β™―β€˜π‘†)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜π‘†))(π‘†β€˜π‘₯) ∈ ran (𝑆 ++ 𝑇)))
7122, 69, 70sylanbrc 584 . . . 4 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ 𝑆:(0..^(β™―β€˜π‘†))⟢ran (𝑆 ++ 𝑇))
7271frnd 6726 . . 3 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ran 𝑆 βŠ† ran (𝑆 ++ 𝑇))
73 ccatval3 14529 . . . . . . . 8 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡 ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜(π‘₯ + (β™―β€˜π‘†))) = (π‘‡β€˜π‘₯))
74733expa 1119 . . . . . . 7 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜(π‘₯ + (β™―β€˜π‘†))) = (π‘‡β€˜π‘₯))
75 elfzouz 13636 . . . . . . . . . 10 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ ∈ (β„€β‰₯β€˜0))
762adantr 482 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (β™―β€˜π‘†) ∈ β„•0)
77 uzaddcl 12888 . . . . . . . . . 10 ((π‘₯ ∈ (β„€β‰₯β€˜0) ∧ (β™―β€˜π‘†) ∈ β„•0) β†’ (π‘₯ + (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜0))
7875, 76, 77syl2anr 598 . . . . . . . . 9 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (π‘₯ + (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜0))
79 nn0addcl 12507 . . . . . . . . . . . 12 (((β™―β€˜π‘†) ∈ β„•0 ∧ (β™―β€˜π‘‡) ∈ β„•0) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ β„•0)
802, 8, 79syl2an 597 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ β„•0)
8180nn0zd 12584 . . . . . . . . . 10 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ β„€)
8281adantr 482 . . . . . . . . 9 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ β„€)
83 elfzonn0 13677 . . . . . . . . . . . 12 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ ∈ β„•0)
8483nn0cnd 12534 . . . . . . . . . . 11 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ ∈ β„‚)
852nn0cnd 12534 . . . . . . . . . . . 12 (𝑆 ∈ Word 𝐡 β†’ (β™―β€˜π‘†) ∈ β„‚)
8685adantr 482 . . . . . . . . . . 11 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (β™―β€˜π‘†) ∈ β„‚)
87 addcom 11400 . . . . . . . . . . 11 ((π‘₯ ∈ β„‚ ∧ (β™―β€˜π‘†) ∈ β„‚) β†’ (π‘₯ + (β™―β€˜π‘†)) = ((β™―β€˜π‘†) + π‘₯))
8884, 86, 87syl2anr 598 . . . . . . . . . 10 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (π‘₯ + (β™―β€˜π‘†)) = ((β™―β€˜π‘†) + π‘₯))
8983nn0red 12533 . . . . . . . . . . . 12 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ ∈ ℝ)
9089adantl 483 . . . . . . . . . . 11 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ ∈ ℝ)
9146ad2antlr 726 . . . . . . . . . . 11 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (β™―β€˜π‘‡) ∈ ℝ)
9244ad2antrr 725 . . . . . . . . . . 11 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (β™―β€˜π‘†) ∈ ℝ)
93 elfzolt2 13641 . . . . . . . . . . . 12 (π‘₯ ∈ (0..^(β™―β€˜π‘‡)) β†’ π‘₯ < (β™―β€˜π‘‡))
9493adantl 483 . . . . . . . . . . 11 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ π‘₯ < (β™―β€˜π‘‡))
9590, 91, 92, 94ltadd2dd 11373 . . . . . . . . . 10 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((β™―β€˜π‘†) + π‘₯) < ((β™―β€˜π‘†) + (β™―β€˜π‘‡)))
9688, 95eqbrtrd 5171 . . . . . . . . 9 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (π‘₯ + (β™―β€˜π‘†)) < ((β™―β€˜π‘†) + (β™―β€˜π‘‡)))
97 elfzo2 13635 . . . . . . . . 9 ((π‘₯ + (β™―β€˜π‘†)) ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) ↔ ((π‘₯ + (β™―β€˜π‘†)) ∈ (β„€β‰₯β€˜0) ∧ ((β™―β€˜π‘†) + (β™―β€˜π‘‡)) ∈ β„€ ∧ (π‘₯ + (β™―β€˜π‘†)) < ((β™―β€˜π‘†) + (β™―β€˜π‘‡))))
9878, 82, 96, 97syl3anbrc 1344 . . . . . . . 8 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (π‘₯ + (β™―β€˜π‘†)) ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))))
99 fnfvelrn 7083 . . . . . . . 8 (((𝑆 ++ 𝑇) Fn (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡))) ∧ (π‘₯ + (β™―β€˜π‘†)) ∈ (0..^((β™―β€˜π‘†) + (β™―β€˜π‘‡)))) β†’ ((𝑆 ++ 𝑇)β€˜(π‘₯ + (β™―β€˜π‘†))) ∈ ran (𝑆 ++ 𝑇))
1001, 98, 99syl2an2r 684 . . . . . . 7 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ ((𝑆 ++ 𝑇)β€˜(π‘₯ + (β™―β€˜π‘†))) ∈ ran (𝑆 ++ 𝑇))
10174, 100eqeltrrd 2835 . . . . . 6 (((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) ∧ π‘₯ ∈ (0..^(β™―β€˜π‘‡))) β†’ (π‘‡β€˜π‘₯) ∈ ran (𝑆 ++ 𝑇))
102101ralrimiva 3147 . . . . 5 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ βˆ€π‘₯ ∈ (0..^(β™―β€˜π‘‡))(π‘‡β€˜π‘₯) ∈ ran (𝑆 ++ 𝑇))
103 ffnfv 7118 . . . . 5 (𝑇:(0..^(β™―β€˜π‘‡))⟢ran (𝑆 ++ 𝑇) ↔ (𝑇 Fn (0..^(β™―β€˜π‘‡)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜π‘‡))(π‘‡β€˜π‘₯) ∈ ran (𝑆 ++ 𝑇)))
10431, 102, 103sylanbrc 584 . . . 4 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ 𝑇:(0..^(β™―β€˜π‘‡))⟢ran (𝑆 ++ 𝑇))
105104frnd 6726 . . 3 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ran 𝑇 βŠ† ran (𝑆 ++ 𝑇))
10672, 105unssd 4187 . 2 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ (ran 𝑆 βˆͺ ran 𝑇) βŠ† ran (𝑆 ++ 𝑇))
10762, 106eqssd 4000 1 ((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ran (𝑆 ++ 𝑇) = (ran 𝑆 βˆͺ ran 𝑇))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βˆͺ cun 3947   βŠ† wss 3949   class class class wbr 5149  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„cr 11109  0cc0 11110   + caddc 11113   < clt 11248   βˆ’ cmin 11444  β„•0cn0 12472  β„€cz 12558  β„€β‰₯cuz 12822  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464   ++ cconcat 14520
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-concat 14521
This theorem is referenced by:  cycpmco2f1  32314  cycpmco2rn  32315  mrsubvrs  34544
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