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Theorem efgs1b 19604
Description: Every extension sequence ending in an irreducible word is trivial. (Contributed by Mario Carneiro, 1-Oct-2015.)
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)))
Assertion
Ref Expression
efgs1b (𝐴 ∈ dom 𝑆 β†’ ((π‘†β€˜π΄) ∈ 𝐷 ↔ (β™―β€˜π΄) = 1))
Distinct variable groups:   𝑦,𝑧   𝑑,𝑛,𝑣,𝑀,𝑦,𝑧,π‘š,π‘₯   π‘š,𝑀   π‘₯,𝑛,𝑀,𝑑,𝑣,𝑀   π‘˜,π‘š,𝑑,π‘₯,𝑇   π‘˜,𝑛,𝑣,𝑀,𝑦,𝑧,π‘Š,π‘š,𝑑,π‘₯   ∼ ,π‘š,𝑑,π‘₯,𝑦,𝑧   π‘š,𝐼,𝑛,𝑑,𝑣,𝑀,π‘₯,𝑦,𝑧   𝐷,π‘š,𝑑
Allowed substitution hints:   𝐴(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 efgs1b
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 eldifn 4128 . . . 4 ((π‘†β€˜π΄) ∈ (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯)) β†’ Β¬ (π‘†β€˜π΄) ∈ βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
2 efgred.d . . . 4 𝐷 = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
31, 2eleq2s 2852 . . 3 ((π‘†β€˜π΄) ∈ 𝐷 β†’ Β¬ (π‘†β€˜π΄) ∈ βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
4 efgval.w . . . . . . . . . 10 π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
5 efgval.r . . . . . . . . . 10 ∼ = ( ~FG β€˜πΌ)
6 efgval2.m . . . . . . . . . 10 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
7 efgval2.t . . . . . . . . . 10 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.s . . . . . . . . . 10 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)))
94, 5, 6, 7, 2, 8efgsdm 19598 . . . . . . . . 9 (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ (π΄β€˜0) ∈ 𝐷 ∧ βˆ€π‘Ž ∈ (1..^(β™―β€˜π΄))(π΄β€˜π‘Ž) ∈ ran (π‘‡β€˜(π΄β€˜(π‘Ž βˆ’ 1)))))
109simp1bi 1146 . . . . . . . 8 (𝐴 ∈ dom 𝑆 β†’ 𝐴 ∈ (Word π‘Š βˆ– {βˆ…}))
11 eldifsn 4791 . . . . . . . . 9 (𝐴 ∈ (Word π‘Š βˆ– {βˆ…}) ↔ (𝐴 ∈ Word π‘Š ∧ 𝐴 β‰  βˆ…))
12 lennncl 14484 . . . . . . . . 9 ((𝐴 ∈ Word π‘Š ∧ 𝐴 β‰  βˆ…) β†’ (β™―β€˜π΄) ∈ β„•)
1311, 12sylbi 216 . . . . . . . 8 (𝐴 ∈ (Word π‘Š βˆ– {βˆ…}) β†’ (β™―β€˜π΄) ∈ β„•)
1410, 13syl 17 . . . . . . 7 (𝐴 ∈ dom 𝑆 β†’ (β™―β€˜π΄) ∈ β„•)
15 elnn1uz2 12909 . . . . . . 7 ((β™―β€˜π΄) ∈ β„• ↔ ((β™―β€˜π΄) = 1 ∨ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)))
1614, 15sylib 217 . . . . . 6 (𝐴 ∈ dom 𝑆 β†’ ((β™―β€˜π΄) = 1 ∨ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)))
1716ord 863 . . . . 5 (𝐴 ∈ dom 𝑆 β†’ (Β¬ (β™―β€˜π΄) = 1 β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)))
1810eldifad 3961 . . . . . . . . . . 11 (𝐴 ∈ dom 𝑆 β†’ 𝐴 ∈ Word π‘Š)
1918adantr 482 . . . . . . . . . 10 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ 𝐴 ∈ Word π‘Š)
20 wrdf 14469 . . . . . . . . . 10 (𝐴 ∈ Word π‘Š β†’ 𝐴:(0..^(β™―β€˜π΄))βŸΆπ‘Š)
2119, 20syl 17 . . . . . . . . 9 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ 𝐴:(0..^(β™―β€˜π΄))βŸΆπ‘Š)
22 1z 12592 . . . . . . . . . . . . . 14 1 ∈ β„€
23 simpr 486 . . . . . . . . . . . . . . 15 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2))
24 df-2 12275 . . . . . . . . . . . . . . . 16 2 = (1 + 1)
2524fveq2i 6895 . . . . . . . . . . . . . . 15 (β„€β‰₯β€˜2) = (β„€β‰₯β€˜(1 + 1))
2623, 25eleqtrdi 2844 . . . . . . . . . . . . . 14 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜(1 + 1)))
27 eluzp1m1 12848 . . . . . . . . . . . . . 14 ((1 ∈ β„€ ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜(1 + 1))) β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ (β„€β‰₯β€˜1))
2822, 26, 27sylancr 588 . . . . . . . . . . . . 13 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ (β„€β‰₯β€˜1))
29 nnuz 12865 . . . . . . . . . . . . 13 β„• = (β„€β‰₯β€˜1)
3028, 29eleqtrrdi 2845 . . . . . . . . . . . 12 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ β„•)
31 lbfzo0 13672 . . . . . . . . . . . 12 (0 ∈ (0..^((β™―β€˜π΄) βˆ’ 1)) ↔ ((β™―β€˜π΄) βˆ’ 1) ∈ β„•)
3230, 31sylibr 233 . . . . . . . . . . 11 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ 0 ∈ (0..^((β™―β€˜π΄) βˆ’ 1)))
33 fzoend 13723 . . . . . . . . . . 11 (0 ∈ (0..^((β™―β€˜π΄) βˆ’ 1)) β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0..^((β™―β€˜π΄) βˆ’ 1)))
34 elfzofz 13648 . . . . . . . . . . 11 ((((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)))
3532, 33, 343syl 18 . . . . . . . . . 10 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0...((β™―β€˜π΄) βˆ’ 1)))
36 eluzelz 12832 . . . . . . . . . . . 12 ((β™―β€˜π΄) ∈ (β„€β‰₯β€˜2) β†’ (β™―β€˜π΄) ∈ β„€)
3736adantl 483 . . . . . . . . . . 11 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (β™―β€˜π΄) ∈ β„€)
38 fzoval 13633 . . . . . . . . . . 11 ((β™―β€˜π΄) ∈ β„€ β†’ (0..^(β™―β€˜π΄)) = (0...((β™―β€˜π΄) βˆ’ 1)))
3937, 38syl 17 . . . . . . . . . 10 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (0..^(β™―β€˜π΄)) = (0...((β™―β€˜π΄) βˆ’ 1)))
4035, 39eleqtrrd 2837 . . . . . . . . 9 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (((β™―β€˜π΄) βˆ’ 1) βˆ’ 1) ∈ (0..^(β™―β€˜π΄)))
4121, 40ffvelcdmd 7088 . . . . . . . 8 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) ∈ π‘Š)
42 uz2m1nn 12907 . . . . . . . . 9 ((β™―β€˜π΄) ∈ (β„€β‰₯β€˜2) β†’ ((β™―β€˜π΄) βˆ’ 1) ∈ β„•)
434, 5, 6, 7, 2, 8efgsdmi 19600 . . . . . . . . 9 ((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))))
4442, 43sylan2 594 . . . . . . . 8 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (π‘†β€˜π΄) ∈ ran (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))
45 fveq2 6892 . . . . . . . . . 10 (π‘Ž = (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) β†’ (π‘‡β€˜π‘Ž) = (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))
4645rneqd 5938 . . . . . . . . 9 (π‘Ž = (π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1)) β†’ ran (π‘‡β€˜π‘Ž) = ran (π‘‡β€˜(π΄β€˜(((β™―β€˜π΄) βˆ’ 1) βˆ’ 1))))
4746eliuni 5004 . . . . . . . 8 (((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†’ (π‘†β€˜π΄) ∈ βˆͺ π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž))
4841, 44, 47syl2anc 585 . . . . . . 7 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (π‘†β€˜π΄) ∈ βˆͺ π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž))
49 fveq2 6892 . . . . . . . . 9 (π‘Ž = π‘₯ β†’ (π‘‡β€˜π‘Ž) = (π‘‡β€˜π‘₯))
5049rneqd 5938 . . . . . . . 8 (π‘Ž = π‘₯ β†’ ran (π‘‡β€˜π‘Ž) = ran (π‘‡β€˜π‘₯))
5150cbviunv 5044 . . . . . . 7 βˆͺ π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) = βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯)
5248, 51eleqtrdi 2844 . . . . . 6 ((𝐴 ∈ dom 𝑆 ∧ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜2)) β†’ (π‘†β€˜π΄) ∈ βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
5352ex 414 . . . . 5 (𝐴 ∈ dom 𝑆 β†’ ((β™―β€˜π΄) ∈ (β„€β‰₯β€˜2) β†’ (π‘†β€˜π΄) ∈ βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯)))
5417, 53syld 47 . . . 4 (𝐴 ∈ dom 𝑆 β†’ (Β¬ (β™―β€˜π΄) = 1 β†’ (π‘†β€˜π΄) ∈ βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯)))
5554con1d 145 . . 3 (𝐴 ∈ dom 𝑆 β†’ (Β¬ (π‘†β€˜π΄) ∈ βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯) β†’ (β™―β€˜π΄) = 1))
563, 55syl5 34 . 2 (𝐴 ∈ dom 𝑆 β†’ ((π‘†β€˜π΄) ∈ 𝐷 β†’ (β™―β€˜π΄) = 1))
579simp2bi 1147 . . . 4 (𝐴 ∈ dom 𝑆 β†’ (π΄β€˜0) ∈ 𝐷)
58 oveq1 7416 . . . . . . 7 ((β™―β€˜π΄) = 1 β†’ ((β™―β€˜π΄) βˆ’ 1) = (1 βˆ’ 1))
59 1m1e0 12284 . . . . . . 7 (1 βˆ’ 1) = 0
6058, 59eqtrdi 2789 . . . . . 6 ((β™―β€˜π΄) = 1 β†’ ((β™―β€˜π΄) βˆ’ 1) = 0)
6160fveq2d 6896 . . . . 5 ((β™―β€˜π΄) = 1 β†’ (π΄β€˜((β™―β€˜π΄) βˆ’ 1)) = (π΄β€˜0))
6261eleq1d 2819 . . . 4 ((β™―β€˜π΄) = 1 β†’ ((π΄β€˜((β™―β€˜π΄) βˆ’ 1)) ∈ 𝐷 ↔ (π΄β€˜0) ∈ 𝐷))
6357, 62syl5ibrcom 246 . . 3 (𝐴 ∈ dom 𝑆 β†’ ((β™―β€˜π΄) = 1 β†’ (π΄β€˜((β™―β€˜π΄) βˆ’ 1)) ∈ 𝐷))
644, 5, 6, 7, 2, 8efgsval 19599 . . . 4 (𝐴 ∈ dom 𝑆 β†’ (π‘†β€˜π΄) = (π΄β€˜((β™―β€˜π΄) βˆ’ 1)))
6564eleq1d 2819 . . 3 (𝐴 ∈ dom 𝑆 β†’ ((π‘†β€˜π΄) ∈ 𝐷 ↔ (π΄β€˜((β™―β€˜π΄) βˆ’ 1)) ∈ 𝐷))
6663, 65sylibrd 259 . 2 (𝐴 ∈ dom 𝑆 β†’ ((β™―β€˜π΄) = 1 β†’ (π‘†β€˜π΄) ∈ 𝐷))
6756, 66impbid 211 1 (𝐴 ∈ dom 𝑆 β†’ ((π‘†β€˜π΄) ∈ 𝐷 ↔ (β™―β€˜π΄) = 1))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  {crab 3433   βˆ– cdif 3946  βˆ…c0 4323  {csn 4629  βŸ¨cop 4635  βŸ¨cotp 4637  βˆͺ ciun 4998   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675  dom cdm 5677  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1oc1o 8459  2oc2o 8460  0cc0 11110  1c1 11111   + caddc 11113   βˆ’ cmin 11444  β„•cn 12212  2c2 12267  β„€cz 12558  β„€β‰₯cuz 12822  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464   splice csplice 14699  βŸ¨β€œcs2 14792   ~FG cefg 19574
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-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465
This theorem is referenced by:  efgredlema  19608  efgredeu  19620
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