Theorem efglem 19645

Description: Lemma for efgval 19646. (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 8725	. 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 6911	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
5	3, 4	eqsstri 3980	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
6	5, 2	sselid 3931	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
7		opelxpi 5661	. . . . . . . . 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 8430	. . . . . . . . . 10 ⊢ (𝑧 ∈ 2o → (1o ∖ 𝑧) ∈ 2o)
10		opelxpi 5661	. . . . . . . . . 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 14794	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩ ∈ Word (𝐼 × 2o))
14		splcl 14675	. . . . . . 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 19644	. . . . . . . 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 2839	. . . . 5 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥))) ∧ (𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩) ∈ 𝑊)
20		brxp 5673	. . . . 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 3179	. . 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 3176	. 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 6848	. . . 4 ⊢ 𝑊 ∈ V
25	24, 24	xpex 7698	. . 3 ⊢ (𝑊 × 𝑊) ∈ V
26		ereq1 8642	. . . 4 ⊢ (𝑟 = (𝑊 × 𝑊) → (𝑟 Er 𝑊 ↔ (𝑊 × 𝑊) Er 𝑊))
27		breq 5100	. . . . . 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 3200	. . . . 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 3200	. . . 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 3560	. 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 2113  ∀wral 3051  Vcvv 3440   ∖ cdif 3898  ⟨cop 4586  ⟨cotp 4588   class class class wbr 5098   I cid 5518   × cxp 5622  ‘cfv 6492  (class class class)co 7358  1_oc1o 8390  2_oc2o 8391   Er wer 8632  0cc0 11026  ...cfz 13423  ♯chash 14253  Word cword 14436   splice csplice 14672  ⟨“cs2 14764

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-hash 14254  df-word 14437  df-concat 14494  df-s1 14520  df-substr 14565  df-pfx 14595  df-splice 14673  df-s2 14771

This theorem is referenced by:  efgval  19646  efger  19647