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Theorem wrdind 14730
Description: Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 12-Oct-2022.)
Hypotheses
Ref Expression
wrdind.1 (𝑥 = ∅ → (𝜑𝜓))
wrdind.2 (𝑥 = 𝑦 → (𝜑𝜒))
wrdind.3 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑𝜃))
wrdind.4 (𝑥 = 𝐴 → (𝜑𝜏))
wrdind.5 𝜓
wrdind.6 ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝜒𝜃))
Assertion
Ref Expression
wrdind (𝐴 ∈ Word 𝐵𝜏)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝑧,𝐵   𝜒,𝑥   𝜑,𝑦,𝑧   𝜏,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑦,𝑧)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)   𝐴(𝑦,𝑧)

Proof of Theorem wrdind
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lencl 14541 . . 3 (𝐴 ∈ Word 𝐵 → (♯‘𝐴) ∈ ℕ0)
2 eqeq2 2738 . . . . . 6 (𝑛 = 0 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 0))
32imbi1d 340 . . . . 5 (𝑛 = 0 → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = 0 → 𝜑)))
43ralbidv 3168 . . . 4 (𝑛 = 0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)))
5 eqeq2 2738 . . . . . 6 (𝑛 = 𝑚 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 𝑚))
65imbi1d 340 . . . . 5 (𝑛 = 𝑚 → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = 𝑚𝜑)))
76ralbidv 3168 . . . 4 (𝑛 = 𝑚 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑)))
8 eqeq2 2738 . . . . . 6 (𝑛 = (𝑚 + 1) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (𝑚 + 1)))
98imbi1d 340 . . . . 5 (𝑛 = (𝑚 + 1) → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
109ralbidv 3168 . . . 4 (𝑛 = (𝑚 + 1) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
11 eqeq2 2738 . . . . . 6 (𝑛 = (♯‘𝐴) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (♯‘𝐴)))
1211imbi1d 340 . . . . 5 (𝑛 = (♯‘𝐴) → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
1312ralbidv 3168 . . . 4 (𝑛 = (♯‘𝐴) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
14 hasheq0 14380 . . . . . 6 (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
15 wrdind.5 . . . . . . 7 𝜓
16 wrdind.1 . . . . . . 7 (𝑥 = ∅ → (𝜑𝜓))
1715, 16mpbiri 257 . . . . . 6 (𝑥 = ∅ → 𝜑)
1814, 17biimtrdi 252 . . . . 5 (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 → 𝜑))
1918rgen 3053 . . . 4 𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)
20 fveqeq2 6910 . . . . . . 7 (𝑥 = 𝑦 → ((♯‘𝑥) = 𝑚 ↔ (♯‘𝑦) = 𝑚))
21 wrdind.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
2220, 21imbi12d 343 . . . . . 6 (𝑥 = 𝑦 → (((♯‘𝑥) = 𝑚𝜑) ↔ ((♯‘𝑦) = 𝑚𝜒)))
2322cbvralvw 3225 . . . . 5 (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑) ↔ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒))
24 simprl 769 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Word 𝐵)
25 fzossfz 13705 . . . . . . . . . . . . . 14 (0..^(♯‘𝑥)) ⊆ (0...(♯‘𝑥))
26 simprr 771 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) = (𝑚 + 1))
27 nn0p1nn 12563 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
2827ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑚 + 1) ∈ ℕ)
2926, 28eqeltrd 2826 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) ∈ ℕ)
30 fzo0end 13778 . . . . . . . . . . . . . . 15 ((♯‘𝑥) ∈ ℕ → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
3129, 30syl 17 . . . . . . . . . . . . . 14 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
3225, 31sselid 3977 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥)))
33 pfxlen 14691 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝐵 ∧ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
3424, 32, 33syl2anc 582 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
3526oveq1d 7439 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) = ((𝑚 + 1) − 1))
36 nn0cn 12534 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
3736ad2antrr 724 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑚 ∈ ℂ)
38 ax-1cn 11216 . . . . . . . . . . . . 13 1 ∈ ℂ
39 pncan 11516 . . . . . . . . . . . . 13 ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
4037, 38, 39sylancl 584 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((𝑚 + 1) − 1) = 𝑚)
4134, 35, 403eqtrd 2770 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚)
42 fveqeq2 6910 . . . . . . . . . . . . 13 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = 𝑚 ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚))
43 vex 3466 . . . . . . . . . . . . . . 15 𝑦 ∈ V
4443, 21sbcie 3820 . . . . . . . . . . . . . 14 ([𝑦 / 𝑥]𝜑𝜒)
45 dfsbcq 3778 . . . . . . . . . . . . . 14 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([𝑦 / 𝑥]𝜑[(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
4644, 45bitr3id 284 . . . . . . . . . . . . 13 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝜒[(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
4742, 46imbi12d 343 . . . . . . . . . . . 12 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (((♯‘𝑦) = 𝑚𝜒) ↔ ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚[(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)))
48 simplr 767 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒))
49 pfxcl 14685 . . . . . . . . . . . . 13 (𝑥 ∈ Word 𝐵 → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵)
5049ad2antrl 726 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵)
5147, 48, 50rspcdva 3609 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚[(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
5241, 51mpd 15 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)
5329nnge1d 12312 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 1 ≤ (♯‘𝑥))
54 wrdlenge1n0 14558 . . . . . . . . . . . . . 14 (𝑥 ∈ Word 𝐵 → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
5554ad2antrl 726 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
5653, 55mpbird 256 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
57 lswcl 14576 . . . . . . . . . . . 12 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → (lastS‘𝑥) ∈ 𝐵)
5824, 56, 57syl2anc 582 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (lastS‘𝑥) ∈ 𝐵)
59 oveq1 7431 . . . . . . . . . . . . . 14 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩))
6059sbceq1d 3781 . . . . . . . . . . . . 13 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
6145, 60imbi12d 343 . . . . . . . . . . . 12 (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (([𝑦 / 𝑥]𝜑[(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
62 s1eq 14608 . . . . . . . . . . . . . . 15 (𝑧 = (lastS‘𝑥) → ⟨“𝑧”⟩ = ⟨“(lastS‘𝑥)”⟩)
6362oveq2d 7440 . . . . . . . . . . . . . 14 (𝑧 = (lastS‘𝑥) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
6463sbceq1d 3781 . . . . . . . . . . . . 13 (𝑧 = (lastS‘𝑥) → ([((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
6564imbi2d 339 . . . . . . . . . . . 12 (𝑧 = (lastS‘𝑥) → (([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑)))
66 wrdind.6 . . . . . . . . . . . . 13 ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝜒𝜃))
67 ovex 7457 . . . . . . . . . . . . . 14 (𝑦 ++ ⟨“𝑧”⟩) ∈ V
68 wrdind.3 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑𝜃))
6967, 68sbcie 3820 . . . . . . . . . . . . 13 ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑𝜃)
7066, 44, 693imtr4g 295 . . . . . . . . . . . 12 ((𝑦 ∈ Word 𝐵𝑧𝐵) → ([𝑦 / 𝑥]𝜑[(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
7161, 65, 70vtocl2ga 3559 . . . . . . . . . . 11 (((𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵 ∧ (lastS‘𝑥) ∈ 𝐵) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
7250, 58, 71syl2anc 582 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
7352, 72mpd 15 . . . . . . . . 9 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑)
74 wrdfin 14540 . . . . . . . . . . . . . 14 (𝑥 ∈ Word 𝐵𝑥 ∈ Fin)
7574ad2antrl 726 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Fin)
76 hashnncl 14383 . . . . . . . . . . . . 13 (𝑥 ∈ Fin → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
7775, 76syl 17 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
7829, 77mpbid 231 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
79 pfxlswccat 14721 . . . . . . . . . . . 12 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) = 𝑥)
8079eqcomd 2732 . . . . . . . . . . 11 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
8124, 78, 80syl2anc 582 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
82 sbceq1a 3787 . . . . . . . . . 10 (𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) → (𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
8381, 82syl 17 . . . . . . . . 9 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝜑[((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
8473, 83mpbird 256 . . . . . . . 8 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝜑)
8584expr 455 . . . . . . 7 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ 𝑥 ∈ Word 𝐵) → ((♯‘𝑥) = (𝑚 + 1) → 𝜑))
8685ralrimiva 3136 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑))
8786ex 411 . . . . 5 (𝑚 ∈ ℕ0 → (∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
8823, 87biimtrid 241 . . . 4 (𝑚 ∈ ℕ0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
894, 7, 10, 13, 19, 88nn0ind 12709 . . 3 ((♯‘𝐴) ∈ ℕ0 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
901, 89syl 17 . 2 (𝐴 ∈ Word 𝐵 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
91 eqidd 2727 . 2 (𝐴 ∈ Word 𝐵 → (♯‘𝐴) = (♯‘𝐴))
92 fveqeq2 6910 . . . 4 (𝑥 = 𝐴 → ((♯‘𝑥) = (♯‘𝐴) ↔ (♯‘𝐴) = (♯‘𝐴)))
93 wrdind.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜏))
9492, 93imbi12d 343 . . 3 (𝑥 = 𝐴 → (((♯‘𝑥) = (♯‘𝐴) → 𝜑) ↔ ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
9594rspcv 3604 . 2 (𝐴 ∈ Word 𝐵 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑) → ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
9690, 91, 95mp2d 49 1 (𝐴 ∈ Word 𝐵𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  [wsbc 3776  c0 4325   class class class wbr 5153  cfv 6554  (class class class)co 7424  Fincfn 8974  cc 11156  0cc0 11158  1c1 11159   + caddc 11161  cle 11299  cmin 11494  cn 12264  0cn0 12524  ...cfz 13538  ..^cfzo 13681  chash 14347  Word cword 14522  lastSclsw 14570   ++ cconcat 14578  ⟨“cs1 14603   prefix cpfx 14678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  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 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-n0 12525  df-xnn0 12597  df-z 12611  df-uz 12875  df-fz 13539  df-fzo 13682  df-hash 14348  df-word 14523  df-lsw 14571  df-concat 14579  df-s1 14604  df-substr 14649  df-pfx 14679
This theorem is referenced by:  frmdgsum  18852  gsumwrev  19363  gsmsymgrfix  19426  efginvrel2  19725  chnind  32880  domnprodn0  33130  unitprodclb  33264  1arithidom  33412  1arithufdlem3  33421  dfufd2lem  33424  signstfvneq0  34418  signstfvc  34420  mrsubvrs  35350
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