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Theorem efgredlemb 19719

Description: The reduced word that forms the base of the sequence in efgsval 19704 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 6834	. . . . . . . . . 10 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → (♯‘(𝑆‘𝐴)) = (♯‘(𝑆‘𝐵)))
10	9	breq2d 5091	. . . . . . . . 9 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) ↔ (♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵))))
11	10	imbi1d 342	. . . . . . . 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 3204	. . . . . . 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 233	. . . . 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 2746	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑆‘𝐴))
18		efgredlem.5	. . . . . 6 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
19		eqcom 2747	. . . . . 6 ⊢ ((𝐴‘0) = (𝐵‘0) ↔ (𝐵‘0) = (𝐴‘0))
20	18, 19	sylnib 329	. . . . 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 2747	. . . . . 6 ⊢ ((𝐴‘𝐾) = (𝐵‘𝐿) ↔ (𝐵‘𝐿) = (𝐴‘𝐾))
31	29, 30	sylnib 329	. . . . 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 19718	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℤ_≥‘𝑃) → (𝐵‘0) = (𝐴‘0)))
33	32, 19	imbitrrdi 253	. . 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 19718	. . 3 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘𝑄) → (𝐴‘0) = (𝐵‘0)))
35	24	elfzelzd 13477	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℤ)
36	23	elfzelzd 13477	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ)
37		uztric 12810	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (𝑄 ∈ (ℤ_≥‘𝑃) ∨ 𝑃 ∈ (ℤ_≥‘𝑄)))
38	35, 36, 37	syl2anc 590	. . 3 ⊢ (𝜑 → (𝑄 ∈ (ℤ_≥‘𝑃) ∨ 𝑃 ∈ (ℤ_≥‘𝑄)))
39	33, 34, 38	mpjaod 866	. 2 ⊢ (𝜑 → (𝐴‘0) = (𝐵‘0))
40	39, 18	pm2.65i 195	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ∀wral 3054 {crab 3392 ∖ cdif 3887 ∅c0 4268 {csn 4562 ⟨cop 4568 ⟨cotp 4570 ∪ ciun 4928 class class class wbr 5079 ↦ cmpt 5160 I cid 5519 × cxp 5623 dom cdm 5625 ran crn 5626 ‘cfv 6492 (class class class)co 7363 ∈ cmpo 7365 1_oc1o 8395 2_oc2o 8396 0cc0 11036 1c1 11037 < clt 11177 − cmin 11375 ℤcz 12522 ℤ_≥cuz 12786 ...cfz 13459 ..^cfzo 13606 ♯chash 14290 Word cword 14473 splice csplice 14709 ⟨“cs2 14801 ~_FG cefg 19679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-ot 4571 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-n0 12436 df-xnn0 12509 df-z 12523 df-uz 12787 df-rp 12941 df-fz 13460 df-fzo 13607 df-hash 14291 df-word 14474 df-concat 14531 df-s1 14557 df-substr 14602 df-pfx 14632 df-splice 14710 df-s2 14808

This theorem is referenced by: efgredlem 19720

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