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Theorem efgredlemb 18473

Description: The reduced word that forms the base of the sequence in efgsval 18457 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-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)))
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)
efgredlemb.p	⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))
efgredlemb.q	⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))
efgredlemb.u	⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_𝑜))
efgredlemb.v	⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_𝑜))
efgredlemb.6	⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
efgredlemb.7	⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
efgredlemb.8	⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))

**Assertion**
Ref	Expression
efgredlemb	⊢ ¬ 𝜑

Distinct variable groups: 𝑎,𝑏,𝐴 𝑦,𝑎,𝑧,𝑏 𝐿,𝑎,𝑏 𝐾,𝑎,𝑏 𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑃 𝑚,𝑎,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑏 𝑈,𝑛,𝑣,𝑤,𝑦,𝑧 𝑘,𝑎,𝑇,𝑏,𝑚,𝑡,𝑥 𝑛,𝑉,𝑣,𝑤,𝑦,𝑧 𝑄,𝑛,𝑡,𝑣,𝑤,𝑦,𝑧 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) 𝑃(𝑥,𝑘,𝑚,𝑎,𝑏) 𝑄(𝑥,𝑘,𝑚,𝑎,𝑏) ∼ (𝑤,𝑣,𝑘,𝑛) 𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑇(𝑦,𝑧,𝑤,𝑣,𝑛) 𝑈(𝑥,𝑡,𝑘,𝑚,𝑎,𝑏) 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) 𝑉(𝑥,𝑡,𝑘,𝑚,𝑎,𝑏)

**Proof of Theorem efgredlemb**
Step	Hyp	Ref	Expression
1		efgval.w	. . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
2		efgval.r	. . . . 5 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . . . 5 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
4		efgval2.t	. . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5		efgred.d	. . . . 5 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
6		efgred.s	. . . . 5 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
7		efgredlem.1	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
8		efgredlem.4	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
9		fveq2 6411	. . . . . . . . . 10 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → (♯‘(𝑆‘𝐴)) = (♯‘(𝑆‘𝐵)))
10	9	breq2d 4855	. . . . . . . . 9 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) ↔ (♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵))))
11	10	imbi1d 333	. . . . . . . 8 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → (((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
12	11	2ralbidv 3170	. . . . . . 7 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
13	8, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
14	7, 13	mpbid 224	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
15		efgredlem.3	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
16		efgredlem.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
17	8	eqcomd 2805	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑆‘𝐴))
18		efgredlem.5	. . . . . 6 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
19		eqcom 2806	. . . . . 6 ⊢ ((𝐴‘0) = (𝐵‘0) ↔ (𝐵‘0) = (𝐴‘0))
20	18, 19	sylnib 320	. . . . 5 ⊢ (𝜑 → ¬ (𝐵‘0) = (𝐴‘0))
21		efgredlemb.l	. . . . 5 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)
22		efgredlemb.k	. . . . 5 ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)
23		efgredlemb.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))
24		efgredlemb.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))
25		efgredlemb.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_𝑜))
26		efgredlemb.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_𝑜))
27		efgredlemb.7	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
28		efgredlemb.6	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
29		efgredlemb.8	. . . . . 6 ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))
30		eqcom 2806	. . . . . 6 ⊢ ((𝐴‘𝐾) = (𝐵‘𝐿) ↔ (𝐵‘𝐿) = (𝐴‘𝐾))
31	29, 30	sylnib 320	. . . . 5 ⊢ (𝜑 → ¬ (𝐵‘𝐿) = (𝐴‘𝐾))
32	1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31	efgredlemc 18472	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℤ_≥‘𝑃) → (𝐵‘0) = (𝐴‘0)))
33	32, 19	syl6ibr 244	. . 3 ⊢ (𝜑 → (𝑄 ∈ (ℤ_≥‘𝑃) → (𝐴‘0) = (𝐵‘0)))
34	1, 2, 3, 4, 5, 6, 7, 16, 15, 8, 18, 22, 21, 24, 23, 26, 25, 28, 27, 29	efgredlemc 18472	. . 3 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘𝑄) → (𝐴‘0) = (𝐵‘0)))
35		elfzelz 12596	. . . . 5 ⊢ (𝑃 ∈ (0...(♯‘(𝐴‘𝐾))) → 𝑃 ∈ ℤ)
36	24, 35	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℤ)
37		elfzelz 12596	. . . . 5 ⊢ (𝑄 ∈ (0...(♯‘(𝐵‘𝐿))) → 𝑄 ∈ ℤ)
38	23, 37	syl 17	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ)
39		uztric 11952	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (𝑄 ∈ (ℤ_≥‘𝑃) ∨ 𝑃 ∈ (ℤ_≥‘𝑄)))
40	36, 38, 39	syl2anc 580	. . 3 ⊢ (𝜑 → (𝑄 ∈ (ℤ_≥‘𝑃) ∨ 𝑃 ∈ (ℤ_≥‘𝑄)))
41	33, 34, 40	mpjaod 887	. 2 ⊢ (𝜑 → (𝐴‘0) = (𝐵‘0))
42	41, 18	pm2.65i 186	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 ∨ wo 874 = wceq 1653 ∈ wcel 2157 ∀wral 3089 {crab 3093 ∖ cdif 3766 ∅c0 4115 {csn 4368 ⟨cop 4374 ⟨cotp 4376 ∪ ciun 4710 class class class wbr 4843 ↦ 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 < clt 10363 − cmin 10556 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-ot 4377 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-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 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-2 11376 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 df-concat 13591 df-s1 13616 df-substr 13665 df-pfx 13714 df-splice 13821 df-s2 13933

This theorem is referenced by: efgredlem 18474 efgredlemOLD 18475

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