Theorem efglem 18569

Description: Lemma for efgval 18570. (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 8221	. 2 ⊢ (𝑊 × 𝑊) Er 𝑊
2		simpll 763	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → 𝑥 ∈ 𝑊)
3		efgval.w	. . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
4		fviss 6611	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
5	3, 4	eqsstri 3924	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
6	5, 2	sseldi 3889	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
7		opelxpi 5483	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2o))
8	7	adantl 482	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → ⟨𝑦, 𝑧⟩ ∈ (𝐼 × 2o))
9		2oconcl 7982	. . . . . . . . . 10 ⊢ (𝑧 ∈ 2o → (1o ∖ 𝑧) ∈ 2o)
10		opelxpi 5483	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐼 ∧ (1o ∖ 𝑧) ∈ 2o) → ⟨𝑦, (1o ∖ 𝑧)⟩ ∈ (𝐼 × 2o))
11	9, 10	sylan2 592	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o) → ⟨𝑦, (1o ∖ 𝑧)⟩ ∈ (𝐼 × 2o))
12	11	adantl 482	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → ⟨𝑦, (1o ∖ 𝑧)⟩ ∈ (𝐼 × 2o))
13	8, 12	s2cld 14069	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩ ∈ Word (𝐼 × 2o))
14		splcl 13950	. . . . . . 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 18568	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
17	16	simprd 496	. . . . . . 7 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
18	17	ad2antrr 722	. . . . . 6 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → 𝑊 = Word (𝐼 × 2o))
19	15, 18	eleqtrrd 2885	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ∈ 𝑊)
20		brxp 5492	. . . . 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 3157	. . 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 3169	. 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 6555	. . . 4 ⊢ 𝑊 ∈ V
25	24, 24	xpex 7336	. . 3 ⊢ (𝑊 × 𝑊) ∈ V
26		ereq1 8149	. . . 4 ⊢ (𝑟 = (𝑊 × 𝑊) → (𝑟 Er 𝑊 ↔ (𝑊 × 𝑊) Er 𝑊))
27		breq 4966	. . . . . 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 3165	. . . . 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 3165	. . . 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 630	. . 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 3547	. 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 688	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 396   = wceq 1522  ∃wex 1762   ∈ wcel 2080  ∀wral 3104  Vcvv 3436   ∖ cdif 3858  ⟨cop 4480  ⟨cotp 4482   class class class wbr 4964   I cid 5350   × cxp 5444  ‘cfv 6228  (class class class)co 7019  1_oc1o 7949  2_oc2o 7950   Er wer 8139  0cc0 10386  ...cfz 12742  ♯chash 13540  Word cword 13707   splice csplice 13947  ⟨“cs2 14039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-ot 4483  df-uni 4748  df-int 4785  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-1st 7548  df-2nd 7549  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-1o 7956  df-2o 7957  df-oadd 7960  df-er 8142  df-map 8261  df-en 8361  df-dom 8362  df-sdom 8363  df-fin 8364  df-card 9217  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-nn 11489  df-n0 11748  df-z 11832  df-uz 12094  df-fz 12743  df-fzo 12884  df-hash 13541  df-word 13708  df-concat 13769  df-s1 13794  df-substr 13839  df-pfx 13869  df-splice 13948  df-s2 14046

This theorem is referenced by:  efgval  18570  efger  18571