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Theorem wrdindOLD 13843
 Description: Obsolete proof of wrdind 13842 as of 12-Oct-2022. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
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
wrdindOLD (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 wrdindOLD
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lencl 13621 . . 3 (𝐴 ∈ Word 𝐵 → (♯‘𝐴) ∈ ℕ0)
2 eqeq2 2789 . . . . . 6 (𝑛 = 0 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 0))
32imbi1d 333 . . . . 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 2789 . . . . . 6 (𝑛 = 𝑚 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 𝑚))
65imbi1d 333 . . . . 5 (𝑛 = 𝑚 → (((♯‘𝑥) = 𝑛𝜑) ↔ ((♯‘𝑥) = 𝑚𝜑)))
76ralbidv 3168 . . . 4 (𝑛 = 𝑚 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑)))
8 eqeq2 2789 . . . . . 6 (𝑛 = (𝑚 + 1) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (𝑚 + 1)))
98imbi1d 333 . . . . 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 2789 . . . . . 6 (𝑛 = (♯‘𝐴) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (♯‘𝐴)))
1211imbi1d 333 . . . . 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 13469 . . . . . 6 (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
15 wrdind.5 . . . . . . 7 𝜓
16 wrdind.1 . . . . . . 7 (𝑥 = ∅ → (𝜑𝜓))
1715, 16mpbiri 250 . . . . . 6 (𝑥 = ∅ → 𝜑)
1814, 17syl6bi 245 . . . . 5 (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 → 𝜑))
1918rgen 3104 . . . 4 𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)
20 fveqeq2 6455 . . . . . . 7 (𝑥 = 𝑦 → ((♯‘𝑥) = 𝑚 ↔ (♯‘𝑦) = 𝑚))
21 wrdind.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
2220, 21imbi12d 336 . . . . . 6 (𝑥 = 𝑦 → (((♯‘𝑥) = 𝑚𝜑) ↔ ((♯‘𝑦) = 𝑚𝜒)))
2322cbvralv 3367 . . . . 5 (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑) ↔ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒))
24 simprl 761 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Word 𝐵)
25 fzossfz 12807 . . . . . . . . . . . . . 14 (0..^(♯‘𝑥)) ⊆ (0...(♯‘𝑥))
26 simprr 763 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) = (𝑚 + 1))
27 nn0p1nn 11683 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
2827ad2antrr 716 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑚 + 1) ∈ ℕ)
2926, 28eqeltrd 2859 . . . . . . . . . . . . . . 15 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) ∈ ℕ)
30 fzo0end 12879 . . . . . . . . . . . . . . 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, 31sseldi 3819 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥)))
33 swrd0lenOLD 13738 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝐵 ∧ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))) → (♯‘(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩)) = ((♯‘𝑥) − 1))
3424, 32, 33syl2anc 579 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩)) = ((♯‘𝑥) − 1))
3526oveq1d 6937 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) = ((𝑚 + 1) − 1))
36 nn0cn 11653 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
3736ad2antrr 716 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑚 ∈ ℂ)
38 ax-1cn 10330 . . . . . . . . . . . . 13 1 ∈ ℂ
39 pncan 10628 . . . . . . . . . . . . 13 ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
4037, 38, 39sylancl 580 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((𝑚 + 1) − 1) = 𝑚)
4134, 35, 403eqtrd 2818 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩)) = 𝑚)
42 fveqeq2 6455 . . . . . . . . . . . . 13 (𝑦 = (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) → ((♯‘𝑦) = 𝑚 ↔ (♯‘(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩)) = 𝑚))
43 vex 3401 . . . . . . . . . . . . . . 15 𝑦 ∈ V
4443, 21sbcie 3687 . . . . . . . . . . . . . 14 ([𝑦 / 𝑥]𝜑𝜒)
45 dfsbcq 3654 . . . . . . . . . . . . . 14 (𝑦 = (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) → ([𝑦 / 𝑥]𝜑[(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑))
4644, 45syl5bbr 277 . . . . . . . . . . . . 13 (𝑦 = (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) → (𝜒[(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑))
4742, 46imbi12d 336 . . . . . . . . . . . 12 (𝑦 = (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) → (((♯‘𝑦) = 𝑚𝜒) ↔ ((♯‘(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩)) = 𝑚[(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑)))
48 simplr 759 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒))
49 swrdcl 13735 . . . . . . . . . . . . 13 (𝑥 ∈ Word 𝐵 → (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ∈ Word 𝐵)
5049ad2antrl 718 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ∈ Word 𝐵)
5147, 48, 50rspcdva 3517 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩)) = 𝑚[(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑))
5241, 51mpd 15 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑)
5329nnge1d 11423 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 1 ≤ (♯‘𝑥))
54 wrdlenge1n0 13640 . . . . . . . . . . . . . 14 (𝑥 ∈ Word 𝐵 → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
5554ad2antrl 718 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
5653, 55mpbird 249 . . . . . . . . . . . 12 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
57 lswcl 13658 . . . . . . . . . . . 12 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → (lastS‘𝑥) ∈ 𝐵)
5824, 56, 57syl2anc 579 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (lastS‘𝑥) ∈ 𝐵)
59 oveq1 6929 . . . . . . . . . . . . . 14 (𝑦 = (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩))
6059sbceq1d 3657 . . . . . . . . . . . . 13 (𝑦 = (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑[((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
6145, 60imbi12d 336 . . . . . . . . . . . 12 (𝑦 = (𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) → (([𝑦 / 𝑥]𝜑[(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑[((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
62 s1eq 13690 . . . . . . . . . . . . . . 15 (𝑧 = (lastS‘𝑥) → ⟨“𝑧”⟩ = ⟨“(lastS‘𝑥)”⟩)
6362oveq2d 6938 . . . . . . . . . . . . . 14 (𝑧 = (lastS‘𝑥) → ((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩))
6463sbceq1d 3657 . . . . . . . . . . . . 13 (𝑧 = (lastS‘𝑥) → ([((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑[((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
6564imbi2d 332 . . . . . . . . . . . 12 (𝑧 = (lastS‘𝑥) → (([(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑[((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑[((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑)))
66 wrdind.6 . . . . . . . . . . . . 13 ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝜒𝜃))
67 ovex 6954 . . . . . . . . . . . . . 14 (𝑦 ++ ⟨“𝑧”⟩) ∈ V
68 wrdind.3 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑𝜃))
6967, 68sbcie 3687 . . . . . . . . . . . . 13 ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑𝜃)
7066, 44, 693imtr4g 288 . . . . . . . . . . . 12 ((𝑦 ∈ Word 𝐵𝑧𝐵) → ([𝑦 / 𝑥]𝜑[(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
7161, 65, 70vtocl2ga 3476 . . . . . . . . . . 11 (((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ∈ Word 𝐵 ∧ (lastS‘𝑥) ∈ 𝐵) → ([(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑[((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
7250, 58, 71syl2anc 579 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ([(𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) / 𝑥]𝜑[((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
7352, 72mpd 15 . . . . . . . . 9 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑)
74 wrdfin 13620 . . . . . . . . . . . . . 14 (𝑥 ∈ Word 𝐵𝑥 ∈ Fin)
7574ad2antrl 718 . . . . . . . . . . . . 13 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Fin)
76 hashnncl 13472 . . . . . . . . . . . . 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 224 . . . . . . . . . . 11 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
79 swrdccatwrdOLD 13830 . . . . . . . . . . . 12 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → ((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩) = 𝑥)
8079eqcomd 2784 . . . . . . . . . . 11 ((𝑥 ∈ Word 𝐵𝑥 ≠ ∅) → 𝑥 = ((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩))
8124, 78, 80syl2anc 579 . . . . . . . . . 10 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 = ((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩))
82 sbceq1a 3663 . . . . . . . . . 10 (𝑥 = ((𝑥 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩) → (𝜑[((𝑥 substr ⟨0, ((♯‘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 substr ⟨0, ((♯‘𝑥) − 1)⟩) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
8473, 83mpbird 249 . . . . . . . 8 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝜑)
8584expr 450 . . . . . . 7 (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) ∧ 𝑥 ∈ Word 𝐵) → ((♯‘𝑥) = (𝑚 + 1) → 𝜑))
8685ralrimiva 3148 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒)) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑))
8786ex 403 . . . . 5 (𝑚 ∈ ℕ0 → (∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚𝜒) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
8823, 87syl5bi 234 . . . 4 (𝑚 ∈ ℕ0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚𝜑) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
894, 7, 10, 13, 19, 88nn0ind 11824 . . 3 ((♯‘𝐴) ∈ ℕ0 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
901, 89syl 17 . 2 (𝐴 ∈ Word 𝐵 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
91 eqidd 2779 . 2 (𝐴 ∈ Word 𝐵 → (♯‘𝐴) = (♯‘𝐴))
92 fveqeq2 6455 . . . 4 (𝑥 = 𝐴 → ((♯‘𝑥) = (♯‘𝐴) ↔ (♯‘𝐴) = (♯‘𝐴)))
93 wrdind.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜏))
9492, 93imbi12d 336 . . 3 (𝑥 = 𝐴 → (((♯‘𝑥) = (♯‘𝐴) → 𝜑) ↔ ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
9594rspcv 3507 . 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 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  [wsbc 3652  ∅c0 4141  ⟨cop 4404   class class class wbr 4886  ‘cfv 6135  (class class class)co 6922  Fincfn 8241  ℂcc 10270  0cc0 10272  1c1 10273   + caddc 10275   ≤ cle 10412   − cmin 10606  ℕcn 11374  ℕ0cn0 11642  ...cfz 12643  ..^cfzo 12784  ♯chash 13435  Word cword 13599  lastSclsw 13652   ++ cconcat 13660  ⟨“cs1 13685   substr csubstr 13730 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-lsw 13653  df-concat 13661  df-s1 13686  df-substr 13731 This theorem is referenced by: (None)
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