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Theorem efgredlemf 19653

Description: Lemma for efgredleme 19655. (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 19642	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1)))))
9	8	simp1bi 1145	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
10	1, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
11	10	eldifad 3909	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Word 𝑊)
12		wrdf 14425	. . . 4 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(♯‘𝐴))⟶𝑊)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))⟶𝑊)
14		fzossfz 13578	. . . . 5 ⊢ (0..^((♯‘𝐴) − 1)) ⊆ (0...((♯‘𝐴) − 1))
15		lencl 14440	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ₀)
16	11, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
17	16	nn0zd 12494	. . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ)
18		fzoval 13560	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℤ → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1)))
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1)))
20	14, 19	sseqtrrid 3973	. . . 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 19652	. . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ))
27	26	simpld 494	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ)
28		fzo0end 13658	. . . . . 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	eqeltrid 2835	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (0..^((♯‘𝐴) − 1)))
31	20, 30	sseldd 3930	. . 3 ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐴)))
32	13, 31	ffvelcdmd 7018	. 2 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
33	2, 3, 4, 5, 6, 7	efgsdm 19642	. . . . . . 7 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1)))))
34	33	simp1bi 1145	. . . . . 6 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
35	23, 34	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
36	35	eldifad 3909	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Word 𝑊)
37		wrdf 14425	. . . 4 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(♯‘𝐵))⟶𝑊)
38	36, 37	syl 17	. . 3 ⊢ (𝜑 → 𝐵:(0..^(♯‘𝐵))⟶𝑊)
39		fzossfz 13578	. . . . 5 ⊢ (0..^((♯‘𝐵) − 1)) ⊆ (0...((♯‘𝐵) − 1))
40		lencl 14440	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑊 → (♯‘𝐵) ∈ ℕ₀)
41	36, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ₀)
42	41	nn0zd 12494	. . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
43		fzoval 13560	. . . . . 6 ⊢ ((♯‘𝐵) ∈ ℤ → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1)))
44	42, 43	syl 17	. . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1)))
45	39, 44	sseqtrrid 3973	. . . 4 ⊢ (𝜑 → (0..^((♯‘𝐵) − 1)) ⊆ (0..^(♯‘𝐵)))
46		efgredlemb.l	. . . . 5 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)
47		fzo0end 13658	. . . . . 6 ⊢ (((♯‘𝐵) − 1) ∈ ℕ → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1)))
48	26, 47	simpl2im 503	. . . . 5 ⊢ (𝜑 → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1)))
49	46, 48	eqeltrid 2835	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (0..^((♯‘𝐵) − 1)))
50	45, 49	sseldd 3930	. . 3 ⊢ (𝜑 → 𝐿 ∈ (0..^(♯‘𝐵)))
51	38, 50	ffvelcdmd 7018	. 2 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
52	32, 51	jca 511	1 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 {crab 3395 ∖ cdif 3894 ∅c0 4280 {csn 4573 ⟨cop 4579 ⟨cotp 4581 ∪ ciun 4939 class class class wbr 5089 ↦ cmpt 5170 I cid 5508 × cxp 5612 dom cdm 5614 ran crn 5615 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1_oc1o 8378 2_oc2o 8379 0cc0 11006 1c1 11007 < clt 11146 − cmin 11344 ℕcn 12125 ℕ₀cn0 12381 ℤcz 12468 ...cfz 13407 ..^cfzo 13554 ♯chash 14237 Word cword 14420 splice csplice 14656 ⟨“cs2 14748 ~_FG cefg 19618

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-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-er 8622 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-2 12188 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-fzo 13555 df-hash 14238 df-word 14421

This theorem is referenced by: efgredlemg 19654 efgredleme 19655

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