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Theorem efgredlemb 19779

Description: The reduced word that forms the base of the sequence in efgsval 19764 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 6907	. . . . . . . . . 10 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → (♯‘(𝑆‘𝐴)) = (♯‘(𝑆‘𝐵)))
10	9	breq2d 5160	. . . . . . . . 9 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) ↔ (♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵))))
11	10	imbi1d 341	. . . . . . . 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 3219	. . . . . . 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 232	. . . . 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 2741	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑆‘𝐴))
18		efgredlem.5	. . . . . 6 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
19		eqcom 2742	. . . . . 6 ⊢ ((𝐴‘0) = (𝐵‘0) ↔ (𝐵‘0) = (𝐴‘0))
20	18, 19	sylnib 328	. . . . 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 2742	. . . . . 6 ⊢ ((𝐴‘𝐾) = (𝐵‘𝐿) ↔ (𝐵‘𝐿) = (𝐴‘𝐾))
31	29, 30	sylnib 328	. . . . 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 19778	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℤ_≥‘𝑃) → (𝐵‘0) = (𝐴‘0)))
33	32, 19	imbitrrdi 252	. . 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 19778	. . 3 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘𝑄) → (𝐴‘0) = (𝐵‘0)))
35	24	elfzelzd 13562	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℤ)
36	23	elfzelzd 13562	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ)
37		uztric 12900	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (𝑄 ∈ (ℤ_≥‘𝑃) ∨ 𝑃 ∈ (ℤ_≥‘𝑄)))
38	35, 36, 37	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑄 ∈ (ℤ_≥‘𝑃) ∨ 𝑃 ∈ (ℤ_≥‘𝑄)))
39	33, 34, 38	mpjaod 860	. 2 ⊢ (𝜑 → (𝐴‘0) = (𝐵‘0))
40	39, 18	pm2.65i 194	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∖ cdif 3960 ∅c0 4339 {csn 4631 ⟨cop 4637 ⟨cotp 4639 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 I cid 5582 × cxp 5687 dom cdm 5689 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1_oc1o 8498 2_oc2o 8499 0cc0 11153 1c1 11154 < clt 11293 − cmin 11490 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Word cword 14549 splice csplice 14784 ⟨“cs2 14877 ~_FG cefg 19739

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-s2 14884

This theorem is referenced by: efgredlem 19780

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