efgcpbl2 - Metamath Proof Explorer

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Theorem efgcpbl2 19694

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 19655	. . 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 8685	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ 𝑊)
7		wrd0 14511	. . . . 5 ⊢ ∅ ∈ Word (𝐼 × 2_o)
8	1	efgrcl 19652	. . . . . . 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 2836	. . . 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 19693	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ ∅ ∈ 𝑊 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
18	6, 11, 12, 17	syl3anc 1373	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
19	6, 10	eleqtrd 2831	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ Word (𝐼 × 2_o))
20	4, 12	ercl 8685	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ 𝑊)
21	20, 10	eleqtrd 2831	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ Word (𝐼 × 2_o))
22		ccatcl 14546	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_o) ∧ 𝐵 ∈ Word (𝐼 × 2_o)) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o))
23	19, 21, 22	syl2anc 584	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o))
24		ccatrid 14559	. . . 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 8687	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ 𝑊)
27	26, 10	eleqtrd 2831	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ Word (𝐼 × 2_o))
28		ccatcl 14546	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_o) ∧ 𝑌 ∈ Word (𝐼 × 2_o)) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o))
29	19, 27, 28	syl2anc 584	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o))
30		ccatrid 14559	. . . 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 5141	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝐴 ++ 𝑌))
33	1, 2, 13, 14, 15, 16	efgcpbl 19693	. . . 4 ⊢ ((∅ ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ 𝐴 ∼ 𝑋) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
34	11, 26, 5, 33	syl3anc 1373	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
35		ccatlid 14558	. . . . 5 ⊢ (𝐴 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝐴) = 𝐴)
36	19, 35	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝐴) = 𝐴)
37	36	oveq1d 7405	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) = (𝐴 ++ 𝑌))
38	4, 5	ercl2 8687	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ 𝑊)
39	38, 10	eleqtrd 2831	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ Word (𝐼 × 2_o))
40		ccatlid 14558	. . . . 5 ⊢ (𝑋 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝑋) = 𝑋)
41	39, 40	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝑋) = 𝑋)
42	41	oveq1d 7405	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝑋) ++ 𝑌) = (𝑋 ++ 𝑌))
43	34, 37, 42	3brtr3d 5141	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∼ (𝑋 ++ 𝑌))
44	4, 32, 43	ertrd 8690	1 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝑋 ++ 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 {crab 3408 Vcvv 3450 ∖ cdif 3914 ∅c0 4299 {csn 4592 ⟨cop 4598 ⟨cotp 4600 ∪ ciun 4958 class class class wbr 5110 ↦ cmpt 5191 I cid 5535 × cxp 5639 ran crn 5642 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 1_oc1o 8430 2_oc2o 8431 Er wer 8671 0cc0 11075 1c1 11076 − cmin 11412 ...cfz 13475 ..^cfzo 13622 ♯chash 14302 Word cword 14485 ++ cconcat 14542 splice csplice 14721 ⟨“cs2 14814 ~_FG cefg 19643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-ot 4601 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-ec 8676 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568 df-substr 14613 df-pfx 14643 df-splice 14722 df-s2 14821 df-efg 19646

This theorem is referenced by: frgpcpbl 19696

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