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Theorem efgredlem 19702
Description: The reduced word that forms the base of the sequence in efgsval 19686 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.) (Proof shortened by AV, 3-Nov-2022.)
Hypotheses
Ref Expression
efgval.w π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
efgval.r ∼ = ( ~FG β€˜πΌ)
efgval2.m 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
efgval2.t 𝑇 = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))
efgred.d 𝐷 = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
efgred.s 𝑆 = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))
efgredlem.1 (πœ‘ β†’ βˆ€π‘Ž ∈ dom π‘†βˆ€π‘ ∈ dom 𝑆((β™―β€˜(π‘†β€˜π‘Ž)) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜π‘Ž) = (π‘†β€˜π‘) β†’ (π‘Žβ€˜0) = (π‘β€˜0))))
efgredlem.2 (πœ‘ β†’ 𝐴 ∈ dom 𝑆)
efgredlem.3 (πœ‘ β†’ 𝐡 ∈ dom 𝑆)
efgredlem.4 (πœ‘ β†’ (π‘†β€˜π΄) = (π‘†β€˜π΅))
efgredlem.5 (πœ‘ β†’ Β¬ (π΄β€˜0) = (π΅β€˜0))
Assertion
Ref Expression
efgredlem Β¬ πœ‘
Distinct variable groups:   π‘Ž,𝑏,𝐴   𝑦,π‘Ž,𝑧,𝑏   𝑑,𝑛,𝑣,𝑀,𝑦,𝑧   π‘š,π‘Ž,𝑛,𝑑,𝑣,𝑀,π‘₯,𝑀,𝑏   π‘˜,π‘Ž,𝑇,𝑏,π‘š,𝑑,π‘₯   π‘Š,π‘Ž,𝑏   π‘˜,𝑛,𝑣,𝑀,𝑦,𝑧,π‘Š,π‘š,𝑑,π‘₯   ∼ ,π‘Ž,𝑏,π‘š,𝑑,π‘₯,𝑦,𝑧   𝐡,π‘Ž,𝑏   𝑆,π‘Ž,𝑏   𝐼,π‘Ž,𝑏,π‘š,𝑛,𝑑,𝑣,𝑀,π‘₯,𝑦,𝑧   𝐷,π‘Ž,𝑏,π‘š,𝑑
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑧,𝑀,𝑣,𝑑,π‘˜,π‘š,𝑛,π‘Ž,𝑏)   𝐴(π‘₯,𝑦,𝑧,𝑀,𝑣,𝑑,π‘˜,π‘š,𝑛)   𝐡(π‘₯,𝑦,𝑧,𝑀,𝑣,𝑑,π‘˜,π‘š,𝑛)   𝐷(π‘₯,𝑦,𝑧,𝑀,𝑣,π‘˜,𝑛)   ∼ (𝑀,𝑣,π‘˜,𝑛)   𝑆(π‘₯,𝑦,𝑧,𝑀,𝑣,𝑑,π‘˜,π‘š,𝑛)   𝑇(𝑦,𝑧,𝑀,𝑣,𝑛)   𝐼(π‘˜)   𝑀(𝑦,𝑧,π‘˜)

Proof of Theorem efgredlem
Dummy variables 𝑖 𝑗 π‘Ÿ 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . . . 10 π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
2 fviss 6975 . . . . . . . . . 10 ( I β€˜Word (𝐼 Γ— 2o)) βŠ† Word (𝐼 Γ— 2o)
31, 2eqsstri 4014 . . . . . . . . 9 π‘Š βŠ† Word (𝐼 Γ— 2o)
4 efgredlem.2 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐴 ∈ dom 𝑆)
5 efgval.r . . . . . . . . . . . . . . 15 ∼ = ( ~FG β€˜πΌ)
6 efgval2.m . . . . . . . . . . . . . . 15 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
7 efgval2.t . . . . . . . . . . . . . . 15 𝑇 = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))
8 efgred.d . . . . . . . . . . . . . . 15 𝐷 = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
9 efgred.s . . . . . . . . . . . . . . 15 𝑆 = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))
101, 5, 6, 7, 8, 9efgsdm 19685 . . . . . . . . . . . . . 14 (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ (π΄β€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜π΄))(π΄β€˜π‘–) ∈ ran (π‘‡β€˜(π΄β€˜(𝑖 βˆ’ 1)))))
1110simp1bi 1143 . . . . . . . . . . . . 13 (𝐴 ∈ dom 𝑆 β†’ 𝐴 ∈ (Word π‘Š βˆ– {βˆ…}))
124, 11syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐴 ∈ (Word π‘Š βˆ– {βˆ…}))
1312eldifad 3959 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ Word π‘Š)
14 wrdf 14502 . . . . . . . . . . 11 (𝐴 ∈ Word π‘Š β†’ 𝐴:(0..^(β™―β€˜π΄))βŸΆπ‘Š)
1513, 14syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐴:(0..^(β™―β€˜π΄))βŸΆπ‘Š)
16 efgredlem.1 . . . . . . . . . . . . . . 15 (πœ‘ β†’ βˆ€π‘Ž ∈ dom π‘†βˆ€π‘ ∈ dom 𝑆((β™―β€˜(π‘†β€˜π‘Ž)) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜π‘Ž) = (π‘†β€˜π‘) β†’ (π‘Žβ€˜0) = (π‘β€˜0))))
17 efgredlem.3 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐡 ∈ dom 𝑆)
18 efgredlem.4 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (π‘†β€˜π΄) = (π‘†β€˜π΅))
19 efgredlem.5 . . . . . . . . . . . . . . 15 (πœ‘ β†’ Β¬ (π΄β€˜0) = (π΅β€˜0))
201, 5, 6, 7, 8, 9, 16, 4, 17, 18, 19efgredlema 19695 . . . . . . . . . . . . . 14 (πœ‘ β†’ (((β™―β€˜π΄) βˆ’ 1) ∈ β„• ∧ ((β™―β€˜π΅) βˆ’ 1) ∈ β„•))
2120simpld 494 . . . . . . . . . . . . 13 (πœ‘ β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ β„•)
22 nnm1nn0 12544 . . . . . . . . . . . . 13 (((β™―β€˜π΄) βˆ’ 1) ∈ β„• β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ β„•0)
2321, 22syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ β„•0)
2421nnred 12258 . . . . . . . . . . . . 13 (πœ‘ β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ ℝ)
2524lem1d 12178 . . . . . . . . . . . 12 (πœ‘ β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ≀ ((β™―β€˜π΄) βˆ’ 1))
26 eldifsni 4794 . . . . . . . . . . . . . . 15 (𝐴 ∈ (Word π‘Š βˆ– {βˆ…}) β†’ 𝐴 β‰  βˆ…)
274, 11, 263syl 18 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐴 β‰  βˆ…)
28 wrdfin 14515 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word π‘Š β†’ 𝐴 ∈ Fin)
29 hashnncl 14358 . . . . . . . . . . . . . . 15 (𝐴 ∈ Fin β†’ ((β™―β€˜π΄) ∈ β„• ↔ 𝐴 β‰  βˆ…))
3013, 28, 293syl 18 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((β™―β€˜π΄) ∈ β„• ↔ 𝐴 β‰  βˆ…))
3127, 30mpbird 257 . . . . . . . . . . . . 13 (πœ‘ β†’ (β™―β€˜π΄) ∈ β„•)
32 nnm1nn0 12544 . . . . . . . . . . . . 13 ((β™―β€˜π΄) ∈ β„• β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ β„•0)
33 fznn0 13626 . . . . . . . . . . . . 13 (((β™―β€˜π΄) βˆ’ 1) ∈ β„•0 β†’ ((((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0...((β™―β€˜π΄) βˆ’ 1)) ↔ ((((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ β„•0 ∧ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ≀ ((β™―β€˜π΄) βˆ’ 1))))
3431, 32, 333syl 18 . . . . . . . . . . . 12 (πœ‘ β†’ ((((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0...((β™―β€˜π΄) βˆ’ 1)) ↔ ((((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ β„•0 ∧ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ≀ ((β™―β€˜π΄) βˆ’ 1))))
3523, 25, 34mpbir2and 712 . . . . . . . . . . 11 (πœ‘ β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0...((β™―β€˜π΄) βˆ’ 1)))
36 lencl 14516 . . . . . . . . . . . . . 14 (𝐴 ∈ Word π‘Š β†’ (β™―β€˜π΄) ∈ β„•0)
3713, 36syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ (β™―β€˜π΄) ∈ β„•0)
3837nn0zd 12615 . . . . . . . . . . . 12 (πœ‘ β†’ (β™―β€˜π΄) ∈ β„€)
39 fzoval 13666 . . . . . . . . . . . 12 ((β™―β€˜π΄) ∈ β„€ β†’ (0..^(β™―β€˜π΄)) = (0...((β™―β€˜π΄) βˆ’ 1)))
4038, 39syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (0..^(β™―β€˜π΄)) = (0...((β™―β€˜π΄) βˆ’ 1)))
4135, 40eleqtrrd 2832 . . . . . . . . . 10 (πœ‘ β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0..^(β™―β€˜π΄)))
4215, 41ffvelcdmd 7095 . . . . . . . . 9 (πœ‘ β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) ∈ π‘Š)
433, 42sselid 3978 . . . . . . . 8 (πœ‘ β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) ∈ Word (𝐼 Γ— 2o))
44 lencl 14516 . . . . . . . 8 ((π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) ∈ Word (𝐼 Γ— 2o) β†’ (β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) ∈ β„•0)
4543, 44syl 17 . . . . . . 7 (πœ‘ β†’ (β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) ∈ β„•0)
4645nn0red 12564 . . . . . 6 (πœ‘ β†’ (β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) ∈ ℝ)
47 2rp 13012 . . . . . 6 2 ∈ ℝ+
48 ltaddrp 13044 . . . . . 6 (((β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) ∈ ℝ ∧ 2 ∈ ℝ+) β†’ (β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) < ((β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) + 2))
4946, 47, 48sylancl 585 . . . . 5 (πœ‘ β†’ (β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) < ((β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) + 2))
5037nn0red 12564 . . . . . . . . . . 11 (πœ‘ β†’ (β™―β€˜π΄) ∈ ℝ)
5150lem1d 12178 . . . . . . . . . 10 (πœ‘ β†’ ((β™―β€˜π΄) βˆ’ 1) ≀ (β™―β€˜π΄))
52 fznn 13602 . . . . . . . . . . 11 ((β™―β€˜π΄) ∈ β„€ β†’ (((β™―β€˜π΄) βˆ’ 1) ∈ (1...(β™―β€˜π΄)) ↔ (((β™―β€˜π΄) βˆ’ 1) ∈ β„• ∧ ((β™―β€˜π΄) βˆ’ 1) ≀ (β™―β€˜π΄))))
5338, 52syl 17 . . . . . . . . . 10 (πœ‘ β†’ (((β™―β€˜π΄) βˆ’ 1) ∈ (1...(β™―β€˜π΄)) ↔ (((β™―β€˜π΄) βˆ’ 1) ∈ β„• ∧ ((β™―β€˜π΄) βˆ’ 1) ≀ (β™―β€˜π΄))))
5421, 51, 53mpbir2and 712 . . . . . . . . 9 (πœ‘ β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ (1...(β™―β€˜π΄)))
551, 5, 6, 7, 8, 9efgsres 19693 . . . . . . . . 9 ((𝐴 ∈ dom 𝑆 ∧ ((β™―β€˜π΄) βˆ’ 1) ∈ (1...(β™―β€˜π΄))) β†’ (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) ∈ dom 𝑆)
564, 54, 55syl2anc 583 . . . . . . . 8 (πœ‘ β†’ (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) ∈ dom 𝑆)
571, 5, 6, 7, 8, 9efgsval 19686 . . . . . . . 8 ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) ∈ dom 𝑆 β†’ (π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜((β™―β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) βˆ’ 1)))
5856, 57syl 17 . . . . . . 7 (πœ‘ β†’ (π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜((β™―β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) βˆ’ 1)))
59 fz1ssfz0 13630 . . . . . . . . . . . 12 (1...(β™―β€˜π΄)) βŠ† (0...(β™―β€˜π΄))
6059, 54sselid 3978 . . . . . . . . . . 11 (πœ‘ β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ (0...(β™―β€˜π΄)))
61 pfxres 14662 . . . . . . . . . . 11 ((𝐴 ∈ Word π‘Š ∧ ((β™―β€˜π΄) βˆ’ 1) ∈ (0...(β™―β€˜π΄))) β†’ (𝐴 prefix ((β™―β€˜π΄) βˆ’ 1)) = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))
6213, 60, 61syl2anc 583 . . . . . . . . . 10 (πœ‘ β†’ (𝐴 prefix ((β™―β€˜π΄) βˆ’ 1)) = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))
6362fveq2d 6901 . . . . . . . . 9 (πœ‘ β†’ (β™―β€˜(𝐴 prefix ((β™―β€˜π΄) βˆ’ 1))) = (β™―β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))))
64 pfxlen 14666 . . . . . . . . . 10 ((𝐴 ∈ Word π‘Š ∧ ((β™―β€˜π΄) βˆ’ 1) ∈ (0...(β™―β€˜π΄))) β†’ (β™―β€˜(𝐴 prefix ((β™―β€˜π΄) βˆ’ 1))) = ((β™―β€˜π΄) βˆ’ 1))
6513, 60, 64syl2anc 583 . . . . . . . . 9 (πœ‘ β†’ (β™―β€˜(𝐴 prefix ((β™―β€˜π΄) βˆ’ 1))) = ((β™―β€˜π΄) βˆ’ 1))
6663, 65eqtr3d 2770 . . . . . . . 8 (πœ‘ β†’ (β™―β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = ((β™―β€˜π΄) βˆ’ 1))
6766fvoveq1d 7442 . . . . . . 7 (πœ‘ β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜((β™―β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) βˆ’ 1)) = ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))
68 fzo0end 13757 . . . . . . . 8 (((β™―β€˜π΄) βˆ’ 1) ∈ β„• β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0..^((β™―β€˜π΄) βˆ’ 1)))
69 fvres 6916 . . . . . . . 8 ((((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0..^((β™―β€˜π΄) βˆ’ 1)) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))
7021, 68, 693syl 18 . . . . . . 7 (πœ‘ β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))
7158, 67, 703eqtrd 2772 . . . . . 6 (πœ‘ β†’ (π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))
7271fveq2d 6901 . . . . 5 (πœ‘ β†’ (β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))) = (β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))
731, 5, 6, 7, 8, 9efgsdmi 19687 . . . . . . 7 ((𝐴 ∈ dom 𝑆 ∧ ((β™―β€˜π΄) βˆ’ 1) ∈ β„•) β†’ (π‘†β€˜π΄) ∈ ran (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))
744, 21, 73syl2anc 583 . . . . . 6 (πœ‘ β†’ (π‘†β€˜π΄) ∈ ran (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))
751, 5, 6, 7efgtlen 19681 . . . . . 6 (((π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) ∈ π‘Š ∧ (π‘†β€˜π΄) ∈ ran (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) β†’ (β™―β€˜(π‘†β€˜π΄)) = ((β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) + 2))
7642, 74, 75syl2anc 583 . . . . 5 (πœ‘ β†’ (β™―β€˜(π‘†β€˜π΄)) = ((β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) + 2))
7749, 72, 763brtr4d 5180 . . . 4 (πœ‘ β†’ (β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))) < (β™―β€˜(π‘†β€˜π΄)))
781, 5, 6, 7efgtf 19677 . . . . . . . . . . . 12 ((π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) ∈ π‘Š β†’ ((π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) = (π‘Ž ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))), 𝑏 ∈ (𝐼 Γ— 2o) ↦ ((π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) splice βŸ¨π‘Ž, π‘Ž, βŸ¨β€œπ‘(π‘€β€˜π‘)β€βŸ©βŸ©)) ∧ (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))):((0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š))
7942, 78syl 17 . . . . . . . . . . 11 (πœ‘ β†’ ((π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) = (π‘Ž ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))), 𝑏 ∈ (𝐼 Γ— 2o) ↦ ((π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) splice βŸ¨π‘Ž, π‘Ž, βŸ¨β€œπ‘(π‘€β€˜π‘)β€βŸ©βŸ©)) ∧ (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))):((0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š))
8079simprd 495 . . . . . . . . . 10 (πœ‘ β†’ (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))):((0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š)
81 ffn 6722 . . . . . . . . . 10 ((π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))):((0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š β†’ (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) Fn ((0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o)))
82 ovelrn 7597 . . . . . . . . . 10 ((π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) Fn ((0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o)) β†’ ((π‘†β€˜π΄) ∈ ran (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) ↔ βˆƒπ‘– ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ)))
8380, 81, 823syl 18 . . . . . . . . 9 (πœ‘ β†’ ((π‘†β€˜π΄) ∈ ran (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))) ↔ βˆƒπ‘– ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ)))
8474, 83mpbid 231 . . . . . . . 8 (πœ‘ β†’ βˆƒπ‘– ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ))
8520simprd 495 . . . . . . . . . 10 (πœ‘ β†’ ((β™―β€˜π΅) βˆ’ 1) ∈ β„•)
861, 5, 6, 7, 8, 9efgsdmi 19687 . . . . . . . . . 10 ((𝐡 ∈ dom 𝑆 ∧ ((β™―β€˜π΅) βˆ’ 1) ∈ β„•) β†’ (π‘†β€˜π΅) ∈ ran (π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))
8717, 85, 86syl2anc 583 . . . . . . . . 9 (πœ‘ β†’ (π‘†β€˜π΅) ∈ ran (π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))
881, 5, 6, 7, 8, 9efgsdm 19685 . . . . . . . . . . . . . . . . 17 (𝐡 ∈ dom 𝑆 ↔ (𝐡 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ (π΅β€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜π΅))(π΅β€˜π‘–) ∈ ran (π‘‡β€˜(π΅β€˜(𝑖 βˆ’ 1)))))
8988simp1bi 1143 . . . . . . . . . . . . . . . 16 (𝐡 ∈ dom 𝑆 β†’ 𝐡 ∈ (Word π‘Š βˆ– {βˆ…}))
9017, 89syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐡 ∈ (Word π‘Š βˆ– {βˆ…}))
9190eldifad 3959 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐡 ∈ Word π‘Š)
92 wrdf 14502 . . . . . . . . . . . . . 14 (𝐡 ∈ Word π‘Š β†’ 𝐡:(0..^(β™―β€˜π΅))βŸΆπ‘Š)
9391, 92syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐡:(0..^(β™―β€˜π΅))βŸΆπ‘Š)
94 fzo0end 13757 . . . . . . . . . . . . . . 15 (((β™―β€˜π΅) βˆ’ 1) ∈ β„• β†’ (((β™―β€˜π΅) βˆ’ 1) βˆ’ 1) ∈ (0..^((β™―β€˜π΅) βˆ’ 1)))
95 elfzofz 13681 . . . . . . . . . . . . . . 15 ((((β™―β€˜π΅) βˆ’ 1) βˆ’ 1) ∈ (0..^((β™―β€˜π΅) βˆ’ 1)) β†’ (((β™―β€˜π΅) βˆ’ 1) βˆ’ 1) ∈ (0...((β™―β€˜π΅) βˆ’ 1)))
9685, 94, 953syl 18 . . . . . . . . . . . . . 14 (πœ‘ β†’ (((β™―β€˜π΅) βˆ’ 1) βˆ’ 1) ∈ (0...((β™―β€˜π΅) βˆ’ 1)))
97 lencl 14516 . . . . . . . . . . . . . . . . 17 (𝐡 ∈ Word π‘Š β†’ (β™―β€˜π΅) ∈ β„•0)
9891, 97syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (β™―β€˜π΅) ∈ β„•0)
9998nn0zd 12615 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (β™―β€˜π΅) ∈ β„€)
100 fzoval 13666 . . . . . . . . . . . . . . 15 ((β™―β€˜π΅) ∈ β„€ β†’ (0..^(β™―β€˜π΅)) = (0...((β™―β€˜π΅) βˆ’ 1)))
10199, 100syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ (0..^(β™―β€˜π΅)) = (0...((β™―β€˜π΅) βˆ’ 1)))
10296, 101eleqtrrd 2832 . . . . . . . . . . . . 13 (πœ‘ β†’ (((β™―β€˜π΅) βˆ’ 1) βˆ’ 1) ∈ (0..^(β™―β€˜π΅)))
10393, 102ffvelcdmd 7095 . . . . . . . . . . . 12 (πœ‘ β†’ (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)) ∈ π‘Š)
1041, 5, 6, 7efgtf 19677 . . . . . . . . . . . 12 ((π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)) ∈ π‘Š β†’ ((π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) = (π‘Ž ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))), 𝑏 ∈ (𝐼 Γ— 2o) ↦ ((π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)) splice βŸ¨π‘Ž, π‘Ž, βŸ¨β€œπ‘(π‘€β€˜π‘)β€βŸ©βŸ©)) ∧ (π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))):((0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š))
105103, 104syl 17 . . . . . . . . . . 11 (πœ‘ β†’ ((π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) = (π‘Ž ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))), 𝑏 ∈ (𝐼 Γ— 2o) ↦ ((π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)) splice βŸ¨π‘Ž, π‘Ž, βŸ¨β€œπ‘(π‘€β€˜π‘)β€βŸ©βŸ©)) ∧ (π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))):((0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š))
106105simprd 495 . . . . . . . . . 10 (πœ‘ β†’ (π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))):((0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š)
107 ffn 6722 . . . . . . . . . 10 ((π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))):((0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š β†’ (π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) Fn ((0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o)))
108 ovelrn 7597 . . . . . . . . . 10 ((π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) Fn ((0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))) Γ— (𝐼 Γ— 2o)) β†’ ((π‘†β€˜π΅) ∈ ran (π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) ↔ βˆƒπ‘— ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))
109106, 107, 1083syl 18 . . . . . . . . 9 (πœ‘ β†’ ((π‘†β€˜π΅) ∈ ran (π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) ↔ βˆƒπ‘— ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))
11087, 109mpbid 231 . . . . . . . 8 (πœ‘ β†’ βˆƒπ‘— ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠))
111 reeanv 3223 . . . . . . . . 9 (βˆƒπ‘– ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))βˆƒπ‘— ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))(βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)) ↔ (βˆƒπ‘– ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ βˆƒπ‘— ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))
112 reeanv 3223 . . . . . . . . . . 11 (βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)βˆƒπ‘  ∈ (𝐼 Γ— 2o)((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)) ↔ (βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))
11316ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ βˆ€π‘Ž ∈ dom π‘†βˆ€π‘ ∈ dom 𝑆((β™―β€˜(π‘†β€˜π‘Ž)) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜π‘Ž) = (π‘†β€˜π‘) β†’ (π‘Žβ€˜0) = (π‘β€˜0))))
1144ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ 𝐴 ∈ dom 𝑆)
11517ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ 𝐡 ∈ dom 𝑆)
11618ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ (π‘†β€˜π΄) = (π‘†β€˜π΅))
11719ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ Β¬ (π΄β€˜0) = (π΅β€˜0))
118 eqid 2728 . . . . . . . . . . . . . . 15 (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) = (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)
119 eqid 2728 . . . . . . . . . . . . . . 15 (((β™―β€˜π΅) βˆ’ 1) βˆ’ 1) = (((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)
120 simpllr 775 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))))
121120simpld 494 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ 𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))))
122120simprd 495 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))
123 simplrl 776 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ (π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)))
124123simpld 494 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ π‘Ÿ ∈ (𝐼 Γ— 2o))
125123simprd 495 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ 𝑠 ∈ (𝐼 Γ— 2o))
126 simplrr 777 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))
127126simpld 494 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ (π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ))
128126simprd 495 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠))
129 simpr 484 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) β†’ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))
1301, 5, 6, 7, 8, 9, 113, 114, 115, 116, 117, 118, 119, 121, 122, 124, 125, 127, 128, 129efgredlemb 19701 . . . . . . . . . . . . . 14 Β¬ (((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))
131 iman 401 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))) ↔ Β¬ (((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) ∧ Β¬ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))
132130, 131mpbir 230 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ ((π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o)) ∧ ((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)))) β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))
133132expr 456 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) ∧ (π‘Ÿ ∈ (𝐼 Γ— 2o) ∧ 𝑠 ∈ (𝐼 Γ— 2o))) β†’ (((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)) β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))
134133rexlimdvva 3208 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) β†’ (βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)βˆƒπ‘  ∈ (𝐼 Γ— 2o)((π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ (π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)) β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))
135112, 134biimtrrid 242 . . . . . . . . . 10 ((πœ‘ ∧ (𝑖 ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))) ∧ 𝑗 ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))))) β†’ ((βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)) β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))
136135rexlimdvva 3208 . . . . . . . . 9 (πœ‘ β†’ (βˆƒπ‘– ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))βˆƒπ‘— ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))(βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)) β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))
137111, 136biimtrrid 242 . . . . . . . 8 (πœ‘ β†’ ((βˆƒπ‘– ∈ (0...(β™―β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))βˆƒπ‘Ÿ ∈ (𝐼 Γ— 2o)(π‘†β€˜π΄) = (𝑖(π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)))π‘Ÿ) ∧ βˆƒπ‘— ∈ (0...(β™―β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))βˆƒπ‘  ∈ (𝐼 Γ— 2o)(π‘†β€˜π΅) = (𝑗(π‘‡β€˜(π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))𝑠)) β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1))))
13884, 110, 137mp2and 698 . . . . . . 7 (πœ‘ β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))
139 fvres 6916 . . . . . . . 8 ((((β™―β€˜π΅) βˆ’ 1) βˆ’ 1) ∈ (0..^((β™―β€˜π΅) βˆ’ 1)) β†’ ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))
14085, 94, 1393syl 18 . . . . . . 7 (πœ‘ β†’ ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)) = (π΅β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))
141138, 70, 1403eqtr4d 2778 . . . . . 6 (πœ‘ β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))
142 fz1ssfz0 13630 . . . . . . . . . . 11 (1...(β™―β€˜π΅)) βŠ† (0...(β™―β€˜π΅))
14398nn0red 12564 . . . . . . . . . . . . 13 (πœ‘ β†’ (β™―β€˜π΅) ∈ ℝ)
144143lem1d 12178 . . . . . . . . . . . 12 (πœ‘ β†’ ((β™―β€˜π΅) βˆ’ 1) ≀ (β™―β€˜π΅))
145 fznn 13602 . . . . . . . . . . . . 13 ((β™―β€˜π΅) ∈ β„€ β†’ (((β™―β€˜π΅) βˆ’ 1) ∈ (1...(β™―β€˜π΅)) ↔ (((β™―β€˜π΅) βˆ’ 1) ∈ β„• ∧ ((β™―β€˜π΅) βˆ’ 1) ≀ (β™―β€˜π΅))))
14699, 145syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (((β™―β€˜π΅) βˆ’ 1) ∈ (1...(β™―β€˜π΅)) ↔ (((β™―β€˜π΅) βˆ’ 1) ∈ β„• ∧ ((β™―β€˜π΅) βˆ’ 1) ≀ (β™―β€˜π΅))))
14785, 144, 146mpbir2and 712 . . . . . . . . . . 11 (πœ‘ β†’ ((β™―β€˜π΅) βˆ’ 1) ∈ (1...(β™―β€˜π΅)))
148142, 147sselid 3978 . . . . . . . . . 10 (πœ‘ β†’ ((β™―β€˜π΅) βˆ’ 1) ∈ (0...(β™―β€˜π΅)))
149 pfxres 14662 . . . . . . . . . 10 ((𝐡 ∈ Word π‘Š ∧ ((β™―β€˜π΅) βˆ’ 1) ∈ (0...(β™―β€˜π΅))) β†’ (𝐡 prefix ((β™―β€˜π΅) βˆ’ 1)) = (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))))
15091, 148, 149syl2anc 583 . . . . . . . . 9 (πœ‘ β†’ (𝐡 prefix ((β™―β€˜π΅) βˆ’ 1)) = (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))))
151150fveq2d 6901 . . . . . . . 8 (πœ‘ β†’ (β™―β€˜(𝐡 prefix ((β™―β€˜π΅) βˆ’ 1))) = (β™―β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))))
152 pfxlen 14666 . . . . . . . . 9 ((𝐡 ∈ Word π‘Š ∧ ((β™―β€˜π΅) βˆ’ 1) ∈ (0...(β™―β€˜π΅))) β†’ (β™―β€˜(𝐡 prefix ((β™―β€˜π΅) βˆ’ 1))) = ((β™―β€˜π΅) βˆ’ 1))
15391, 148, 152syl2anc 583 . . . . . . . 8 (πœ‘ β†’ (β™―β€˜(𝐡 prefix ((β™―β€˜π΅) βˆ’ 1))) = ((β™―β€˜π΅) βˆ’ 1))
154151, 153eqtr3d 2770 . . . . . . 7 (πœ‘ β†’ (β™―β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) = ((β™―β€˜π΅) βˆ’ 1))
155154fvoveq1d 7442 . . . . . 6 (πœ‘ β†’ ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜((β™―β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) βˆ’ 1)) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜(((β™―β€˜π΅) βˆ’ 1) βˆ’ 1)))
156141, 67, 1553eqtr4d 2778 . . . . 5 (πœ‘ β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜((β™―β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) βˆ’ 1)) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜((β™―β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) βˆ’ 1)))
1571, 5, 6, 7, 8, 9efgsres 19693 . . . . . . 7 ((𝐡 ∈ dom 𝑆 ∧ ((β™―β€˜π΅) βˆ’ 1) ∈ (1...(β™―β€˜π΅))) β†’ (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) ∈ dom 𝑆)
15817, 147, 157syl2anc 583 . . . . . 6 (πœ‘ β†’ (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) ∈ dom 𝑆)
1591, 5, 6, 7, 8, 9efgsval 19686 . . . . . 6 ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) ∈ dom 𝑆 β†’ (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜((β™―β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) βˆ’ 1)))
160158, 159syl 17 . . . . 5 (πœ‘ β†’ (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜((β™―β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) βˆ’ 1)))
161156, 58, 1603eqtr4d 2778 . . . 4 (πœ‘ β†’ (π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))))
162 fveq2 6897 . . . . . . . . 9 (π‘Ž = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) β†’ (π‘†β€˜π‘Ž) = (π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))))
163162fveq2d 6901 . . . . . . . 8 (π‘Ž = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) β†’ (β™―β€˜(π‘†β€˜π‘Ž)) = (β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))))
164163breq1d 5158 . . . . . . 7 (π‘Ž = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) β†’ ((β™―β€˜(π‘†β€˜π‘Ž)) < (β™―β€˜(π‘†β€˜π΄)) ↔ (β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))) < (β™―β€˜(π‘†β€˜π΄))))
165162eqeq1d 2730 . . . . . . . 8 (π‘Ž = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) β†’ ((π‘†β€˜π‘Ž) = (π‘†β€˜π‘) ↔ (π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜π‘)))
166 fveq1 6896 . . . . . . . . 9 (π‘Ž = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) β†’ (π‘Žβ€˜0) = ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0))
167166eqeq1d 2730 . . . . . . . 8 (π‘Ž = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) β†’ ((π‘Žβ€˜0) = (π‘β€˜0) ↔ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = (π‘β€˜0)))
168165, 167imbi12d 344 . . . . . . 7 (π‘Ž = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) β†’ (((π‘†β€˜π‘Ž) = (π‘†β€˜π‘) β†’ (π‘Žβ€˜0) = (π‘β€˜0)) ↔ ((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜π‘) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = (π‘β€˜0))))
169164, 168imbi12d 344 . . . . . 6 (π‘Ž = (𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) β†’ (((β™―β€˜(π‘†β€˜π‘Ž)) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜π‘Ž) = (π‘†β€˜π‘) β†’ (π‘Žβ€˜0) = (π‘β€˜0))) ↔ ((β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜π‘) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = (π‘β€˜0)))))
170 fveq2 6897 . . . . . . . . 9 (𝑏 = (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) β†’ (π‘†β€˜π‘) = (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))))
171170eqeq2d 2739 . . . . . . . 8 (𝑏 = (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) β†’ ((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜π‘) ↔ (π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))))))
172 fveq1 6896 . . . . . . . . 9 (𝑏 = (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) β†’ (π‘β€˜0) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜0))
173172eqeq2d 2739 . . . . . . . 8 (𝑏 = (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) β†’ (((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = (π‘β€˜0) ↔ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜0)))
174171, 173imbi12d 344 . . . . . . 7 (𝑏 = (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) β†’ (((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜π‘) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = (π‘β€˜0)) ↔ ((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜0))))
175174imbi2d 340 . . . . . 6 (𝑏 = (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) β†’ (((β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜π‘) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = (π‘β€˜0))) ↔ ((β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜0)))))
176169, 175rspc2va 3621 . . . . 5 ((((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))) ∈ dom 𝑆 ∧ (𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1))) ∈ dom 𝑆) ∧ βˆ€π‘Ž ∈ dom π‘†βˆ€π‘ ∈ dom 𝑆((β™―β€˜(π‘†β€˜π‘Ž)) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜π‘Ž) = (π‘†β€˜π‘) β†’ (π‘Žβ€˜0) = (π‘β€˜0)))) β†’ ((β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜0))))
17756, 158, 16, 176syl21anc 837 . . . 4 (πœ‘ β†’ ((β™―β€˜(π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1))))) < (β™―β€˜(π‘†β€˜π΄)) β†’ ((π‘†β€˜(𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))) = (π‘†β€˜(𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))) β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜0))))
17877, 161, 177mp2d 49 . . 3 (πœ‘ β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜0))
179 lbfzo0 13705 . . . . 5 (0 ∈ (0..^((β™―β€˜π΄) βˆ’ 1)) ↔ ((β™―β€˜π΄) βˆ’ 1) ∈ β„•)
18021, 179sylibr 233 . . . 4 (πœ‘ β†’ 0 ∈ (0..^((β™―β€˜π΄) βˆ’ 1)))
181180fvresd 6917 . . 3 (πœ‘ β†’ ((𝐴 β†Ύ (0..^((β™―β€˜π΄) βˆ’ 1)))β€˜0) = (π΄β€˜0))
182 lbfzo0 13705 . . . . 5 (0 ∈ (0..^((β™―β€˜π΅) βˆ’ 1)) ↔ ((β™―β€˜π΅) βˆ’ 1) ∈ β„•)
18385, 182sylibr 233 . . . 4 (πœ‘ β†’ 0 ∈ (0..^((β™―β€˜π΅) βˆ’ 1)))
184183fvresd 6917 . . 3 (πœ‘ β†’ ((𝐡 β†Ύ (0..^((β™―β€˜π΅) βˆ’ 1)))β€˜0) = (π΅β€˜0))
185178, 181, 1843eqtr3d 2776 . 2 (πœ‘ β†’ (π΄β€˜0) = (π΅β€˜0))
186185, 19pm2.65i 193 1 Β¬ πœ‘
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067  {crab 3429   βˆ– cdif 3944  βˆ…c0 4323  {csn 4629  βŸ¨cop 4635  βŸ¨cotp 4637  βˆͺ ciun 4996   class class class wbr 5148   ↦ cmpt 5231   I cid 5575   Γ— cxp 5676  dom cdm 5678  ran crn 5679   β†Ύ cres 5680   Fn wfn 6543  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420   ∈ cmpo 7422  1oc1o 8480  2oc2o 8481  Fincfn 8964  β„cr 11138  0cc0 11139  1c1 11140   + caddc 11142   < clt 11279   ≀ cle 11280   βˆ’ cmin 11475  β„•cn 12243  2c2 12298  β„•0cn0 12503  β„€cz 12589  β„+crp 13007  ...cfz 13517  ..^cfzo 13660  β™―chash 14322  Word cword 14497   prefix cpfx 14653   splice csplice 14732  βŸ¨β€œcs2 14825   ~FG cefg 19661
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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-n0 12504  df-xnn0 12576  df-z 12590  df-uz 12854  df-rp 13008  df-fz 13518  df-fzo 13661  df-hash 14323  df-word 14498  df-concat 14554  df-s1 14579  df-substr 14624  df-pfx 14654  df-splice 14733  df-s2 14832
This theorem is referenced by:  efgred  19703
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