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Theorem efglem 18325
Description: Lemma for efgval 18326. (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 7981 . 2 (𝑊 × 𝑊) Er 𝑊
2 simpll 807 . . . . 5 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝑥𝑊)
3 efgval.w . . . . . . . . 9 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
4 fviss 6414 . . . . . . . . 9 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
53, 4eqsstri 3772 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2𝑜)
65, 2sseldi 3738 . . . . . . 7 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
7 opelxpi 5301 . . . . . . . . 9 ((𝑦𝐼𝑧 ∈ 2𝑜) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2𝑜))
87adantl 473 . . . . . . . 8 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2𝑜))
9 2oconcl 7748 . . . . . . . . . 10 (𝑧 ∈ 2𝑜 → (1𝑜𝑧) ∈ 2𝑜)
10 opelxpi 5301 . . . . . . . . . 10 ((𝑦𝐼 ∧ (1𝑜𝑧) ∈ 2𝑜) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
119, 10sylan2 492 . . . . . . . . 9 ((𝑦𝐼𝑧 ∈ 2𝑜) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
1211adantl 473 . . . . . . . 8 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
138, 12s2cld 13812 . . . . . . 7 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩ ∈ Word (𝐼 × 2𝑜))
14 splcl 13699 . . . . . . 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 696 . . . . . 6 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩) ∈ Word (𝐼 × 2𝑜))
163efgrcl 18324 . . . . . . . 8 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
1716simprd 482 . . . . . . 7 (𝑥𝑊𝑊 = Word (𝐼 × 2𝑜))
1817ad2antrr 764 . . . . . 6 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝑊 = Word (𝐼 × 2𝑜))
1915, 18eleqtrrd 2838 . . . . 5 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩) ∈ 𝑊)
20 brxp 5300 . . . . 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 701 . . . 4 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))
2221ralrimivva 3105 . . 3 ((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) → ∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))
2322rgen2 3109 . 2 𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)
24 fvex 6358 . . . . 5 ( I ‘Word (𝐼 × 2𝑜)) ∈ V
253, 24eqeltri 2831 . . . 4 𝑊 ∈ V
2625, 25xpex 7123 . . 3 (𝑊 × 𝑊) ∈ V
27 ereq1 7914 . . . 4 (𝑟 = (𝑊 × 𝑊) → (𝑟 Er 𝑊 ↔ (𝑊 × 𝑊) Er 𝑊))
28 breq 4802 . . . . . 6 (𝑟 = (𝑊 × 𝑊) → (𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩) ↔ 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)))
29282ralbidv 3123 . . . . 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)⟩”⟩⟩)))
30292ralbidv 3123 . . . 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)⟩”⟩⟩)))
3127, 30anbi12d 749 . . 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)⟩”⟩⟩))))
3226, 31spcev 3436 . 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)⟩”⟩⟩)))
331, 23, 32mp2an 710 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 383   = wceq 1628  wex 1849  wcel 2135  wral 3046  Vcvv 3336  cdif 3708  cop 4323  cotp 4325   class class class wbr 4800   I cid 5169   × cxp 5260  cfv 6045  (class class class)co 6809  1𝑜c1o 7718  2𝑜c2o 7719   Er wer 7904  0cc0 10124  ...cfz 12515  chash 13307  Word cword 13473   splice csplice 13478  ⟨“cs2 13782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-ot 4326  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-2o 7726  df-oadd 7729  df-er 7907  df-map 8021  df-pm 8022  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-card 8951  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-n0 11481  df-z 11566  df-uz 11876  df-fz 12516  df-fzo 12656  df-hash 13308  df-word 13481  df-concat 13483  df-s1 13484  df-substr 13485  df-splice 13486  df-s2 13789
This theorem is referenced by:  efgval  18326  efger  18327
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