efgcpbl2 - Metamath Proof Explorer

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Theorem efgcpbl2 19278

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 19239	. . 3 ⊢ ∼ Er 𝑊
4	3	a1i 11	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ∼ Er 𝑊)
5		simpl 482	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∼ 𝑋)
6	4, 5	ercl 8467	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ 𝑊)
7		wrd0 14170	. . . . 5 ⊢ ∅ ∈ Word (𝐼 × 2_o)
8	1	efgrcl 19236	. . . . . . 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 495	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑊 = Word (𝐼 × 2_o))
11	7, 10	eleqtrrid 2846	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ∅ ∈ 𝑊)
12		simpr 484	. . . 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 19277	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ ∅ ∈ 𝑊 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
18	6, 11, 12, 17	syl3anc 1369	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
19	6, 10	eleqtrd 2841	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ Word (𝐼 × 2_o))
20	4, 12	ercl 8467	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ 𝑊)
21	20, 10	eleqtrd 2841	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ Word (𝐼 × 2_o))
22		ccatcl 14205	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_o) ∧ 𝐵 ∈ Word (𝐼 × 2_o)) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o))
23	19, 21, 22	syl2anc 583	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o))
24		ccatrid 14220	. . . 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 8469	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ 𝑊)
27	26, 10	eleqtrd 2841	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ Word (𝐼 × 2_o))
28		ccatcl 14205	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_o) ∧ 𝑌 ∈ Word (𝐼 × 2_o)) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o))
29	19, 27, 28	syl2anc 583	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o))
30		ccatrid 14220	. . . 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 5101	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝐴 ++ 𝑌))
33	1, 2, 13, 14, 15, 16	efgcpbl 19277	. . . 4 ⊢ ((∅ ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ 𝐴 ∼ 𝑋) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
34	11, 26, 5, 33	syl3anc 1369	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
35		ccatlid 14219	. . . . 5 ⊢ (𝐴 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝐴) = 𝐴)
36	19, 35	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝐴) = 𝐴)
37	36	oveq1d 7270	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) = (𝐴 ++ 𝑌))
38	4, 5	ercl2 8469	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ 𝑊)
39	38, 10	eleqtrd 2841	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ Word (𝐼 × 2_o))
40		ccatlid 14219	. . . . 5 ⊢ (𝑋 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝑋) = 𝑋)
41	39, 40	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝑋) = 𝑋)
42	41	oveq1d 7270	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝑋) ++ 𝑌) = (𝑋 ++ 𝑌))
43	34, 37, 42	3brtr3d 5101	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∼ (𝑋 ++ 𝑌))
44	4, 32, 43	ertrd 8472	1 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝑋 ++ 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 Vcvv 3422 ∖ cdif 3880 ∅c0 4253 {csn 4558 ⟨cop 4564 ⟨cotp 4566 ∪ ciun 4921 class class class wbr 5070 ↦ cmpt 5153 I cid 5479 × cxp 5578 ran crn 5581 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 1_oc1o 8260 2_oc2o 8261 Er wer 8453 0cc0 10802 1c1 10803 − cmin 11135 ...cfz 13168 ..^cfzo 13311 ♯chash 13972 Word cword 14145 ++ cconcat 14201 splice csplice 14390 ⟨“cs2 14482 ~_FG cefg 19227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-ec 8458 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-splice 14391 df-s2 14489 df-efg 19230

This theorem is referenced by: frgpcpbl 19280

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