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Theorem efglem 18838
 Description: Lemma for efgval 18839. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypothesis
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2o))
Assertion
Ref Expression
efglem 𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))
Distinct variable groups:   𝑦,𝑟,𝑧,𝑛,𝑥,𝑊   𝑛,𝐼,𝑟,𝑥,𝑦,𝑧

Proof of Theorem efglem
StepHypRef Expression
1 xpider 8355 . 2 (𝑊 × 𝑊) Er 𝑊
2 simpll 766 . . . . 5 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → 𝑥𝑊)
3 efgval.w . . . . . . . . 9 𝑊 = ( I ‘Word (𝐼 × 2o))
4 fviss 6720 . . . . . . . . 9 ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
53, 4eqsstri 3952 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2o)
65, 2sseldi 3916 . . . . . . 7 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
7 opelxpi 5560 . . . . . . . . 9 ((𝑦𝐼𝑧 ∈ 2o) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2o))
87adantl 485 . . . . . . . 8 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2o))
9 2oconcl 8115 . . . . . . . . . 10 (𝑧 ∈ 2o → (1o𝑧) ∈ 2o)
10 opelxpi 5560 . . . . . . . . . 10 ((𝑦𝐼 ∧ (1o𝑧) ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
119, 10sylan2 595 . . . . . . . . 9 ((𝑦𝐼𝑧 ∈ 2o) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
1211adantl 485 . . . . . . . 8 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → ⟨𝑦, (1o𝑧)⟩ ∈ (𝐼 × 2o))
138, 12s2cld 14228 . . . . . . 7 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩ ∈ Word (𝐼 × 2o))
14 splcl 14109 . . . . . . 7 ((𝑥 ∈ Word (𝐼 × 2o) ∧ ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩ ∈ Word (𝐼 × 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩) ∈ Word (𝐼 × 2o))
156, 13, 14syl2anc 587 . . . . . 6 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩) ∈ Word (𝐼 × 2o))
163efgrcl 18837 . . . . . . . 8 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
1716simprd 499 . . . . . . 7 (𝑥𝑊𝑊 = Word (𝐼 × 2o))
1817ad2antrr 725 . . . . . 6 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → 𝑊 = Word (𝐼 × 2o))
1915, 18eleqtrrd 2896 . . . . 5 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩) ∈ 𝑊)
20 brxp 5569 . . . . 5 (𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩) ↔ (𝑥𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩) ∈ 𝑊))
212, 19, 20sylanbrc 586 . . . 4 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2o)) → 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))
2221ralrimivva 3159 . . 3 ((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) → ∀𝑦𝐼𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))
2322rgen2 3171 . 2 𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)
243fvexi 6663 . . . 4 𝑊 ∈ V
2524, 24xpex 7460 . . 3 (𝑊 × 𝑊) ∈ V
26 ereq1 8283 . . . 4 (𝑟 = (𝑊 × 𝑊) → (𝑟 Er 𝑊 ↔ (𝑊 × 𝑊) Er 𝑊))
27 breq 5035 . . . . . 6 (𝑟 = (𝑊 × 𝑊) → (𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩) ↔ 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)))
28272ralbidv 3167 . . . . 5 (𝑟 = (𝑊 × 𝑊) → (∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩) ↔ ∀𝑦𝐼𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)))
29282ralbidv 3167 . . . 4 (𝑟 = (𝑊 × 𝑊) → (∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩) ↔ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)))
3026, 29anbi12d 633 . . 3 (𝑟 = (𝑊 × 𝑊) → ((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)) ↔ ((𝑊 × 𝑊) Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))))
3125, 30spcev 3558 . 2 (((𝑊 × 𝑊) Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)) → ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩)))
321, 23, 31mp2an 691 1 𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o𝑧)⟩”⟩⟩))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  ∀wral 3109  Vcvv 3444   ∖ cdif 3881  ⟨cop 4534  ⟨cotp 4536   class class class wbr 5033   I cid 5427   × cxp 5521  ‘cfv 6328  (class class class)co 7139  1oc1o 8082  2oc2o 8083   Er wer 8273  0cc0 10530  ...cfz 12889  ♯chash 13690  Word cword 13861   splice csplice 14106  ⟨“cs2 14198 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-ot 4537  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028  df-splice 14107  df-s2 14205 This theorem is referenced by:  efgval  18839  efger  18840
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