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Theorem efgredlemf 17977

Description: Lemma for efgredleme 17979. (Contributed by Mario Carneiro, 4-Jun-2016.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
efgred.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
efgred.s	⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
efgredlem.1	⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
efgredlem.2	⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
efgredlem.3	⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
efgredlem.4	⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
efgredlem.5	⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
efgredlemb.k	⊢ 𝐾 = (((#‘𝐴) − 1) − 1)
efgredlemb.l	⊢ 𝐿 = (((#‘𝐵) − 1) − 1)

**Assertion**
Ref	Expression
efgredlemf	⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊))

Distinct variable groups: 𝑎,𝑏,𝐴 𝑦,𝑎,𝑧,𝑏 𝐿,𝑎,𝑏 𝐾,𝑎,𝑏 𝑡,𝑛,𝑣,𝑤,𝑦,𝑧 𝑚,𝑎,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑏 𝑘,𝑎,𝑇,𝑏,𝑚,𝑡,𝑥 𝑊,𝑎,𝑏 𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥 ∼ ,𝑎,𝑏,𝑚,𝑡,𝑥,𝑦,𝑧 𝐵,𝑎,𝑏 𝑆,𝑎,𝑏 𝐼,𝑎,𝑏,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝐷,𝑎,𝑏,𝑚,𝑡 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛,𝑎,𝑏) 𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐵(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛) ∼ (𝑤,𝑣,𝑘,𝑛) 𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑇(𝑦,𝑧,𝑤,𝑣,𝑛) 𝐼(𝑘) 𝐾(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑀(𝑦,𝑧,𝑘)

Proof of Theorem efgredlemf Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgredlem.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
2		efgval.w	. . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
3		efgval.r	. . . . . . . 8 ⊢ ∼ = ( ~_FG ‘𝐼)
4		efgval2.m	. . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
5		efgval2.t	. . . . . . . 8 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
6		efgred.d	. . . . . . . 8 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
7		efgred.s	. . . . . . . 8 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
8	2, 3, 4, 5, 6, 7	efgsdm 17966	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1)))))
9	8	simp1bi 1069	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
10	1, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
11	10	eldifad 3552	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Word 𝑊)
12		wrdf 13165	. . . 4 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(#‘𝐴))⟶𝑊)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → 𝐴:(0..^(#‘𝐴))⟶𝑊)
14		fzossfz 12357	. . . . 5 ⊢ (0..^((#‘𝐴) − 1)) ⊆ (0...((#‘𝐴) − 1))
15		lencl 13179	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (#‘𝐴) ∈ ℕ₀)
16	11, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘𝐴) ∈ ℕ₀)
17	16	nn0zd 11356	. . . . . 6 ⊢ (𝜑 → (#‘𝐴) ∈ ℤ)
18		fzoval 12340	. . . . . 6 ⊢ ((#‘𝐴) ∈ ℤ → (0..^(#‘𝐴)) = (0...((#‘𝐴) − 1)))
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → (0..^(#‘𝐴)) = (0...((#‘𝐴) − 1)))
20	14, 19	syl5sseqr 3617	. . . 4 ⊢ (𝜑 → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
21		efgredlemb.k	. . . . 5 ⊢ 𝐾 = (((#‘𝐴) − 1) − 1)
22		efgredlem.1	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
23		efgredlem.3	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
24		efgredlem.4	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
25		efgredlem.5	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
26	2, 3, 4, 5, 6, 7, 22, 1, 23, 24, 25	efgredlema 17976	. . . . . . 7 ⊢ (𝜑 → (((#‘𝐴) − 1) ∈ ℕ ∧ ((#‘𝐵) − 1) ∈ ℕ))
27	26	simpld 474	. . . . . 6 ⊢ (𝜑 → ((#‘𝐴) − 1) ∈ ℕ)
28		fzo0end 12426	. . . . . 6 ⊢ (((#‘𝐴) − 1) ∈ ℕ → (((#‘𝐴) − 1) − 1) ∈ (0..^((#‘𝐴) − 1)))
29	27, 28	syl 17	. . . . 5 ⊢ (𝜑 → (((#‘𝐴) − 1) − 1) ∈ (0..^((#‘𝐴) − 1)))
30	21, 29	syl5eqel 2692	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (0..^((#‘𝐴) − 1)))
31	20, 30	sseldd 3569	. . 3 ⊢ (𝜑 → 𝐾 ∈ (0..^(#‘𝐴)))
32	13, 31	ffvelrnd 6268	. 2 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
33	2, 3, 4, 5, 6, 7	efgsdm 17966	. . . . . . 7 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1)))))
34	33	simp1bi 1069	. . . . . 6 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
35	23, 34	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
36	35	eldifad 3552	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Word 𝑊)
37		wrdf 13165	. . . 4 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(#‘𝐵))⟶𝑊)
38	36, 37	syl 17	. . 3 ⊢ (𝜑 → 𝐵:(0..^(#‘𝐵))⟶𝑊)
39		fzossfz 12357	. . . . 5 ⊢ (0..^((#‘𝐵) − 1)) ⊆ (0...((#‘𝐵) − 1))
40		lencl 13179	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑊 → (#‘𝐵) ∈ ℕ₀)
41	36, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ₀)
42	41	nn0zd 11356	. . . . . 6 ⊢ (𝜑 → (#‘𝐵) ∈ ℤ)
43		fzoval 12340	. . . . . 6 ⊢ ((#‘𝐵) ∈ ℤ → (0..^(#‘𝐵)) = (0...((#‘𝐵) − 1)))
44	42, 43	syl 17	. . . . 5 ⊢ (𝜑 → (0..^(#‘𝐵)) = (0...((#‘𝐵) − 1)))
45	39, 44	syl5sseqr 3617	. . . 4 ⊢ (𝜑 → (0..^((#‘𝐵) − 1)) ⊆ (0..^(#‘𝐵)))
46		efgredlemb.l	. . . . 5 ⊢ 𝐿 = (((#‘𝐵) − 1) − 1)
47	26	simprd 478	. . . . . 6 ⊢ (𝜑 → ((#‘𝐵) − 1) ∈ ℕ)
48		fzo0end 12426	. . . . . 6 ⊢ (((#‘𝐵) − 1) ∈ ℕ → (((#‘𝐵) − 1) − 1) ∈ (0..^((#‘𝐵) − 1)))
49	47, 48	syl 17	. . . . 5 ⊢ (𝜑 → (((#‘𝐵) − 1) − 1) ∈ (0..^((#‘𝐵) − 1)))
50	46, 49	syl5eqel 2692	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (0..^((#‘𝐵) − 1)))
51	45, 50	sseldd 3569	. . 3 ⊢ (𝜑 → 𝐿 ∈ (0..^(#‘𝐵)))
52	38, 51	ffvelrnd 6268	. 2 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
53	32, 52	jca 553	1 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∅c0 3874 {csn 4125 ⟨cop 4131 ⟨cotp 4133 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1_𝑜c1o 7440 2_𝑜c2o 7441 0cc0 9815 1c1 9816 < clt 9953 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 splice csplice 13151 ⟨“cs2 13437 ~_FG cefg 17942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154

This theorem is referenced by: efgredlemg 17978 efgredleme 17979

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