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Theorem efglem 18336
Description: Lemma for efgval 18337. (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 7974 . 2 (𝑊 × 𝑊) Er 𝑊
2 simpll 750 . . . . 5 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝑥𝑊)
3 efgval.w . . . . . . . . 9 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
4 fviss 6400 . . . . . . . . 9 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
53, 4eqsstri 3784 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2𝑜)
65, 2sseldi 3750 . . . . . . 7 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
7 opelxpi 5287 . . . . . . . . 9 ((𝑦𝐼𝑧 ∈ 2𝑜) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2𝑜))
87adantl 467 . . . . . . . 8 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2𝑜))
9 2oconcl 7741 . . . . . . . . . 10 (𝑧 ∈ 2𝑜 → (1𝑜𝑧) ∈ 2𝑜)
10 opelxpi 5287 . . . . . . . . . 10 ((𝑦𝐼 ∧ (1𝑜𝑧) ∈ 2𝑜) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
119, 10sylan2 580 . . . . . . . . 9 ((𝑦𝐼𝑧 ∈ 2𝑜) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
1211adantl 467 . . . . . . . 8 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → ⟨𝑦, (1𝑜𝑧)⟩ ∈ (𝐼 × 2𝑜))
138, 12s2cld 13825 . . . . . . 7 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩ ∈ Word (𝐼 × 2𝑜))
14 splcl 13712 . . . . . . 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 573 . . . . . 6 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩) ∈ Word (𝐼 × 2𝑜))
163efgrcl 18335 . . . . . . . 8 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
1716simprd 483 . . . . . . 7 (𝑥𝑊𝑊 = Word (𝐼 × 2𝑜))
1817ad2antrr 705 . . . . . 6 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝑊 = Word (𝐼 × 2𝑜))
1915, 18eleqtrrd 2853 . . . . 5 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩) ∈ 𝑊)
20 brxp 5286 . . . . 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 572 . . . 4 (((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦𝐼𝑧 ∈ 2𝑜)) → 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))
2221ralrimivva 3120 . . 3 ((𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))) → ∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))
2322rgen2 3124 . 2 𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩)
243fvexi 6345 . . . 4 𝑊 ∈ V
2524, 24xpex 7113 . . 3 (𝑊 × 𝑊) ∈ V
26 ereq1 7907 . . . 4 (𝑟 = (𝑊 × 𝑊) → (𝑟 Er 𝑊 ↔ (𝑊 × 𝑊) Er 𝑊))
27 breq 4789 . . . . . 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 3138 . . . . 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 3138 . . . 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 616 . . 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 3451 . 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 672 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 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  Vcvv 3351  cdif 3720  cop 4323  cotp 4325   class class class wbr 4787   I cid 5157   × cxp 5248  cfv 6030  (class class class)co 6796  1𝑜c1o 7710  2𝑜c2o 7711   Er wer 7897  0cc0 10142  ...cfz 12533  chash 13321  Word cword 13487   splice csplice 13492  ⟨“cs2 13795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-ot 4326  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-n0 11500  df-z 11585  df-uz 11894  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-substr 13499  df-splice 13500  df-s2 13802
This theorem is referenced by:  efgval  18337  efger  18338
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