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Theorem efgsdmi 18458

Description: Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-2015.)

**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)))

**Assertion**
Ref	Expression
efgsdmi	⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) ∈ ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1))))

Distinct variable groups: 𝑦,𝑧 𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥 𝑚,𝑀 𝑥,𝑛,𝑀,𝑡,𝑣,𝑤 𝑘,𝑚,𝑡,𝑥,𝑇 𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥 ∼ ,𝑚,𝑡,𝑥,𝑦,𝑧 𝑚,𝐼,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝐷,𝑚,𝑡 Allowed substitution hints: 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) 𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐼(𝑘) 𝑀(𝑦,𝑧,𝑘)

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

Step	Hyp	Ref	Expression
1		efgval.w	. . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
2		efgval.r	. . . 4 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . . 4 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
4		efgval2.t	. . . 4 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5		efgred.d	. . . 4 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
6		efgred.s	. . . 4 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
7	1, 2, 3, 4, 5, 6	efgsval 18457	. . 3 ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) = (𝐹‘((♯‘𝐹) − 1)))
8	7	adantr 473	. 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) = (𝐹‘((♯‘𝐹) − 1)))
9		fveq2 6411	. . . 4 ⊢ (𝑖 = ((♯‘𝐹) − 1) → (𝐹‘𝑖) = (𝐹‘((♯‘𝐹) − 1)))
10		fvoveq1 6901	. . . . . 6 ⊢ (𝑖 = ((♯‘𝐹) − 1) → (𝐹‘(𝑖 − 1)) = (𝐹‘(((♯‘𝐹) − 1) − 1)))
11	10	fveq2d 6415	. . . . 5 ⊢ (𝑖 = ((♯‘𝐹) − 1) → (𝑇‘(𝐹‘(𝑖 − 1))) = (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1))))
12	11	rneqd 5556	. . . 4 ⊢ (𝑖 = ((♯‘𝐹) − 1) → ran (𝑇‘(𝐹‘(𝑖 − 1))) = ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1))))
13	9, 12	eleq12d 2872	. . 3 ⊢ (𝑖 = ((♯‘𝐹) − 1) → ((𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))) ↔ (𝐹‘((♯‘𝐹) − 1)) ∈ ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1)))))
14	1, 2, 3, 4, 5, 6	efgsdm 18456	. . . . 5 ⊢ (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))
15	14	simp3bi 1178	. . . 4 ⊢ (𝐹 ∈ dom 𝑆 → ∀𝑖 ∈ (1..^(♯‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))
16	15	adantr 473	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → ∀𝑖 ∈ (1..^(♯‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))
17		simpr 478	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → ((♯‘𝐹) − 1) ∈ ℕ)
18		nnuz 11967	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
19	17, 18	syl6eleq 2888	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → ((♯‘𝐹) − 1) ∈ (ℤ_≥‘1))
20		eluzfz1 12602	. . . . . 6 ⊢ (((♯‘𝐹) − 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...((♯‘𝐹) − 1)))
21	19, 20	syl 17	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → 1 ∈ (1...((♯‘𝐹) − 1)))
22	14	simp1bi 1176	. . . . . . . 8 ⊢ (𝐹 ∈ dom 𝑆 → 𝐹 ∈ (Word 𝑊 ∖ {∅}))
23	22	adantr 473	. . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → 𝐹 ∈ (Word 𝑊 ∖ {∅}))
24	23	eldifad 3781	. . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → 𝐹 ∈ Word 𝑊)
25		lencl 13553	. . . . . 6 ⊢ (𝐹 ∈ Word 𝑊 → (♯‘𝐹) ∈ ℕ₀)
26		nn0z 11690	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℤ)
27		fzoval 12726	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℤ → (1..^(♯‘𝐹)) = (1...((♯‘𝐹) − 1)))
28	24, 25, 26, 27	4syl 19	. . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (1..^(♯‘𝐹)) = (1...((♯‘𝐹) − 1)))
29	21, 28	eleqtrrd 2881	. . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → 1 ∈ (1..^(♯‘𝐹)))
30		fzoend 12814	. . . 4 ⊢ (1 ∈ (1..^(♯‘𝐹)) → ((♯‘𝐹) − 1) ∈ (1..^(♯‘𝐹)))
31	29, 30	syl 17	. . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → ((♯‘𝐹) − 1) ∈ (1..^(♯‘𝐹)))
32	13, 16, 31	rspcdva 3503	. 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (𝐹‘((♯‘𝐹) − 1)) ∈ ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1))))
33	8, 32	eqeltrd 2878	1 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) ∈ ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 {crab 3093 ∖ cdif 3766 ∅c0 4115 {csn 4368 ⟨cop 4374 ⟨cotp 4376 ∪ ciun 4710 ↦ cmpt 4922 I cid 5219 × cxp 5310 dom cdm 5312 ran crn 5313 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 1_𝑜c1o 7792 2_𝑜c2o 7793 0cc0 10224 1c1 10225 − cmin 10556 ℕcn 11312 ℕ₀cn0 11580 ℤcz 11666 ℤ_≥cuz 11930 ...cfz 12580 ..^cfzo 12720 ♯chash 13370 Word cword 13534 splice csplice 13819 ⟨“cs2 13926 ~_FG cefg 18432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535

This theorem is referenced by: efgs1b 18462 efgredlemg 18469 efgredlemd 18471 efgredlem 18474 efgredlemOLD 18475

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