Theorem efglem 18129

Description: Lemma for efgval 18130. (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 7818	. 2 ⊢ (𝑊 × 𝑊) Er 𝑊
2		simpll 790	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → 𝑥 ∈ 𝑊)
3		efgval.w	. . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
4		fviss 6256	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
5	3, 4	eqsstri 3635	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2𝑜)
6	5, 2	sseldi 3601	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
7		opelxpi 5148	. . . . . . . . 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 7583	. . . . . . . . . 10 ⊢ (𝑧 ∈ 2𝑜 → (1𝑜 ∖ 𝑧) ∈ 2𝑜)
10		opelxpi 5148	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐼 ∧ (1𝑜 ∖ 𝑧) ∈ 2𝑜) → ⟨𝑦, (1𝑜 ∖ 𝑧)⟩ ∈ (𝐼 × 2𝑜))
11	9, 10	sylan2 491	. . . . . . . . 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 13616	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩ ∈ Word (𝐼 × 2𝑜))
14		splcl 13503	. . . . . . 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 693	. . . . . 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 18128	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
17	16	simprd 479	. . . . . . 7 ⊢ (𝑥 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2𝑜))
18	17	ad2antrr 762	. . . . . 6 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → 𝑊 = Word (𝐼 × 2𝑜))
19	15, 18	eleqtrrd 2704	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(#‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩) ∈ 𝑊)
20		brxp 5147	. . . . 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 698	. . . 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 2971	. . 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 2975	. 2 ⊢ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 𝑥(𝑊 × 𝑊)(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜 ∖ 𝑧)⟩”⟩⟩)
24		fvex 6201	. . . . 5 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ∈ V
25	3, 24	eqeltri 2697	. . . 4 ⊢ 𝑊 ∈ V
26	25, 25	xpex 6962	. . 3 ⊢ (𝑊 × 𝑊) ∈ V
27		ereq1 7749	. . . 4 ⊢ (𝑟 = (𝑊 × 𝑊) → (𝑟 Er 𝑊 ↔ (𝑊 × 𝑊) Er 𝑊))
28		breq 4655	. . . . . 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)⟩”⟩⟩)))
29	28	2ralbidv 2989	. . . . 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)⟩”⟩⟩)))
30	29	2ralbidv 2989	. . . 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)⟩”⟩⟩)))
31	27, 30	anbi12d 747	. . 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)⟩”⟩⟩))))
32	26, 31	spcev 3300	. 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)⟩”⟩⟩)))
33	1, 23, 32	mp2an 708	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 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571  ⟨cop 4183  ⟨cotp 4185   class class class wbr 4653   I cid 5023   × cxp 5112  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   Er wer 7739  0cc0 9936  ...cfz 12326  #chash 13117  Word cword 13291   splice csplice 13296  ⟨“cs2 13586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593

This theorem is referenced by:  efgval  18130  efger  18131