efgcpbl2 - Metamath Proof Explorer

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Theorem efgcpbl2 18524

Description: Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-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
efgcpbl2	⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝑋 ++ 𝑌))

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: 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) 𝑀(𝑦,𝑧,𝑘) 𝑋(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑌(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)

**Proof of Theorem efgcpbl2**
Step	Hyp	Ref	Expression
1		efgval.w	. . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
2		efgval.r	. . . 4 ⊢ ∼ = ( ~_FG ‘𝐼)
3	1, 2	efger 18483	. . 3 ⊢ ∼ Er 𝑊
4	3	a1i 11	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ∼ Er 𝑊)
5		simpl 476	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∼ 𝑋)
6	4, 5	ercl 8021	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ 𝑊)
7		wrd0 13600	. . . . 5 ⊢ ∅ ∈ Word (𝐼 × 2_o)
8	1	efgrcl 18480	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_o)))
9	6, 8	syl 17	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_o)))
10	9	simprd 491	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑊 = Word (𝐼 × 2_o))
11	7, 10	syl5eleqr 2914	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ∅ ∈ 𝑊)
12		simpr 479	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∼ 𝑌)
13		efgval2.m	. . . . 5 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
14		efgval2.t	. . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
15		efgred.d	. . . . 5 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
16		efgred.s	. . . . 5 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
17	1, 2, 13, 14, 15, 16	efgcpbl 18523	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ ∅ ∈ 𝑊 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
18	6, 11, 12, 17	syl3anc 1496	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
19	6, 10	eleqtrd 2909	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ Word (𝐼 × 2_o))
20	4, 12	ercl 8021	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ 𝑊)
21	20, 10	eleqtrd 2909	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ Word (𝐼 × 2_o))
22		ccatcl 13635	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_o) ∧ 𝐵 ∈ Word (𝐼 × 2_o)) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o))
23	19, 21, 22	syl2anc 581	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o))
24		ccatrid 13648	. . . 4 ⊢ ((𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o) → ((𝐴 ++ 𝐵) ++ ∅) = (𝐴 ++ 𝐵))
25	23, 24	syl 17	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) = (𝐴 ++ 𝐵))
26	4, 12	ercl2 8023	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ 𝑊)
27	26, 10	eleqtrd 2909	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ Word (𝐼 × 2_o))
28		ccatcl 13635	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_o) ∧ 𝑌 ∈ Word (𝐼 × 2_o)) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o))
29	19, 27, 28	syl2anc 581	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o))
30		ccatrid 13648	. . . 4 ⊢ ((𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o) → ((𝐴 ++ 𝑌) ++ ∅) = (𝐴 ++ 𝑌))
31	29, 30	syl 17	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝑌) ++ ∅) = (𝐴 ++ 𝑌))
32	18, 25, 31	3brtr3d 4905	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝐴 ++ 𝑌))
33	1, 2, 13, 14, 15, 16	efgcpbl 18523	. . . 4 ⊢ ((∅ ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ 𝐴 ∼ 𝑋) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
34	11, 26, 5, 33	syl3anc 1496	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
35		ccatlid 13647	. . . . 5 ⊢ (𝐴 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝐴) = 𝐴)
36	19, 35	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝐴) = 𝐴)
37	36	oveq1d 6921	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) = (𝐴 ++ 𝑌))
38	4, 5	ercl2 8023	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ 𝑊)
39	38, 10	eleqtrd 2909	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ Word (𝐼 × 2_o))
40		ccatlid 13647	. . . . 5 ⊢ (𝑋 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝑋) = 𝑋)
41	39, 40	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝑋) = 𝑋)
42	41	oveq1d 6921	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝑋) ++ 𝑌) = (𝑋 ++ 𝑌))
43	34, 37, 42	3brtr3d 4905	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∼ (𝑋 ++ 𝑌))
44	4, 32, 43	ertrd 8026	1 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝑋 ++ 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 {crab 3122 Vcvv 3415 ∖ cdif 3796 ∅c0 4145 {csn 4398 ⟨cop 4404 ⟨cotp 4406 ∪ ciun 4741 class class class wbr 4874 ↦ cmpt 4953 I cid 5250 × cxp 5341 ran crn 5344 ‘cfv 6124 (class class class)co 6906 ↦ cmpt2 6908 1_oc1o 7820 2_oc2o 7821 Er wer 8007 0cc0 10253 1c1 10254 − cmin 10586 ...cfz 12620 ..^cfzo 12761 ♯chash 13411 Word cword 13575 ++ cconcat 13631 splice csplice 13856 ⟨“cs2 13963 ~_FG cefg 18471

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-2o 7828 df-oadd 7831 df-er 8010 df-ec 8012 df-map 8125 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-fzo 12762 df-hash 13412 df-word 13576 df-concat 13632 df-s1 13657 df-substr 13702 df-pfx 13751 df-splice 13858 df-s2 13970 df-efg 18474

This theorem is referenced by: frgpcpbl 18526

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