Theorem efglem 19628

Description: Lemma for efgval 19629. (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 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))

Distinct variable groups:   𝑦,𝑟,𝑧,𝑛,𝑥,𝑊   𝑛,𝐼,𝑟,𝑥,𝑦,𝑧

Proof of Theorem efglem

Step	Hyp	Ref	Expression
1		xpider 8712	. 2 ⊢ (𝑊 × 𝑊) Er 𝑊
2		simpll 766	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → 𝑥 ∈ 𝑊)
3		efgval.w	. . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
4		fviss 6899	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
5	3, 4	eqsstri 3976	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
6	5, 2	sselid 3927	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
7		opelxpi 5651	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2o))
8	7	adantl 481	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2o))
9		2oconcl 8418	. . . . . . . . . 10 ⊢ (𝑧 ∈ 2o → (1o ∖ 𝑧) ∈ 2o)
10		opelxpi 5651	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐼 ∧ (1o ∖ 𝑧) ∈ 2o) → ⟨𝑦, (1o ∖ 𝑧)⟩ ∈ (𝐼 × 2o))
11	9, 10	sylan2 593	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o) → ⟨𝑦, (1o ∖ 𝑧)⟩ ∈ (𝐼 × 2o))
12	11	adantl 481	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → ⟨𝑦, (1o ∖ 𝑧)⟩ ∈ (𝐼 × 2o))
13	8, 12	s2cld 14778	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩ ∈ Word (𝐼 × 2o))
14		splcl 14659	. . . . . . 7 ⊢ ((𝑥 ∈ Word (𝐼 × 2o) ∧ ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩ ∈ Word (𝐼 × 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ∈ Word (𝐼 × 2o))
15	6, 13, 14	syl2anc 584	. . . . . 6 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ∈ Word (𝐼 × 2o))
16	3	efgrcl 19627	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
17	16	simprd 495	. . . . . . 7 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
18	17	ad2antrr 726	. . . . . 6 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → 𝑊 = Word (𝐼 × 2o))
19	15, 18	eleqtrrd 2834	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ∈ 𝑊)
20		brxp 5663	. . . . 5 ⊢ (𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ↔ (𝑥 ∈ 𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ∈ 𝑊))
21	2, 19, 20	sylanbrc 583	. . . 4 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))
22	21	ralrimivva 3175	. . 3 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) → ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))
23	22	rgen2 3172	. 2 ⊢ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩)
24	3	fvexi 6836	. . . 4 ⊢ 𝑊 ∈ V
25	24, 24	xpex 7686	. . 3 ⊢ (𝑊 × 𝑊) ∈ V
26		ereq1 8629	. . . 4 ⊢ (𝑟 = (𝑊 × 𝑊) → (𝑟 Er 𝑊 ↔ (𝑊 × 𝑊) Er 𝑊))
27		breq 5091	. . . . . 6 ⊢ (𝑟 = (𝑊 × 𝑊) → (𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ↔ 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩)))
28	27	2ralbidv 3196	. . . . 5 ⊢ (𝑟 = (𝑊 × 𝑊) → (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ↔ ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩)))
29	28	2ralbidv 3196	. . . 4 ⊢ (𝑟 = (𝑊 × 𝑊) → (∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ↔ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩)))
30	26, 29	anbi12d 632	. . 3 ⊢ (𝑟 = (𝑊 × 𝑊) → ((𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩)) ↔ ((𝑊 × 𝑊) Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))))
31	25, 30	spcev 3556	. 2 ⊢ (((𝑊 × 𝑊) Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩)) → ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩)))
32	1, 23, 31	mp2an 692	1 ⊢ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∖ cdif 3894  ⟨cop 4579  ⟨cotp 4581   class class class wbr 5089   I cid 5508   × cxp 5612  ‘cfv 6481  (class class class)co 7346  1_oc1o 8378  2_oc2o 8379   Er wer 8619  0cc0 11006  ...cfz 13407  ♯chash 14237  Word cword 14420   splice csplice 14656  ⟨“cs2 14748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-hash 14238  df-word 14421  df-concat 14478  df-s1 14504  df-substr 14549  df-pfx 14579  df-splice 14657  df-s2 14755

This theorem is referenced by:  efgval  19629  efger  19630