efgcpbl2 - Metamath Proof Explorer

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Theorem efgcpbl2 18383

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 18344	. . 3 ⊢ ∼ Er 𝑊
4	3	a1i 11	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ∼ Er 𝑊)
5		simpl 475	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∼ 𝑋)
6	4, 5	ercl 7905	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ 𝑊)
7		wrd0 13529	. . . . 5 ⊢ ∅ ∈ Word (𝐼 × 2_𝑜)
8	1	efgrcl 18341	. . . . . . 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 483	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑊 = Word (𝐼 × 2_𝑜))
11	7, 10	syl5eleqr 2855	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ∅ ∈ 𝑊)
12		simpr 480	. . . 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 18382	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ ∅ ∈ 𝑊 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
18	6, 11, 12, 17	syl3anc 1474	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
19	6, 10	eleqtrd 2850	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ Word (𝐼 × 2_𝑜))
20	4, 12	ercl 7905	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ 𝑊)
21	20, 10	eleqtrd 2850	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ Word (𝐼 × 2_𝑜))
22		ccatcl 13559	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_𝑜) ∧ 𝐵 ∈ Word (𝐼 × 2_𝑜)) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_𝑜))
23	19, 21, 22	syl2anc 693	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_𝑜))
24		ccatrid 13572	. . . 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 7907	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ 𝑊)
27	26, 10	eleqtrd 2850	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ Word (𝐼 × 2_𝑜))
28		ccatcl 13559	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑌 ∈ Word (𝐼 × 2_𝑜)) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_𝑜))
29	19, 27, 28	syl2anc 693	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_𝑜))
30		ccatrid 13572	. . . 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 4814	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝐴 ++ 𝑌))
33	1, 2, 13, 14, 15, 16	efgcpbl 18382	. . . 4 ⊢ ((∅ ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ 𝐴 ∼ 𝑋) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
34	11, 26, 5, 33	syl3anc 1474	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
35		ccatlid 13571	. . . . 5 ⊢ (𝐴 ∈ Word (𝐼 × 2_𝑜) → (∅ ++ 𝐴) = 𝐴)
36	19, 35	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝐴) = 𝐴)
37	36	oveq1d 6806	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) = (𝐴 ++ 𝑌))
38	4, 5	ercl2 7907	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ 𝑊)
39	38, 10	eleqtrd 2850	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ Word (𝐼 × 2_𝑜))
40		ccatlid 13571	. . . . 5 ⊢ (𝑋 ∈ Word (𝐼 × 2_𝑜) → (∅ ++ 𝑋) = 𝑋)
41	39, 40	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝑋) = 𝑋)
42	41	oveq1d 6806	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝑋) ++ 𝑌) = (𝑋 ++ 𝑌))
43	34, 37, 42	3brtr3d 4814	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∼ (𝑋 ++ 𝑌))
44	4, 32, 43	ertrd 7910	1 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝑋 ++ 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1629 ∈ wcel 2143 ∀wral 3059 {crab 3063 Vcvv 3348 ∖ cdif 3717 ∅c0 4060 {csn 4313 ⟨cop 4319 ⟨cotp 4321 ∪ ciun 4651 class class class wbr 4783 ↦ cmpt 4860 I cid 5155 × cxp 5246 ran crn 5249 ‘cfv 6030 (class class class)co 6791 ↦ cmpt2 6793 1_𝑜c1o 7704 2_𝑜c2o 7705 Er wer 7891 0cc0 10136 1c1 10137 − cmin 10466 ...cfz 12532 ..^cfzo 12672 ♯chash 13324 Word cword 13490 ++ cconcat 13492 splice csplice 13495 ⟨“cs2 13798 ~_FG cefg 18332

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1868 ax-4 1883 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2145 ax-9 2152 ax-10 2172 ax-11 2188 ax-12 2201 ax-13 2406 ax-ext 2749 ax-rep 4901 ax-sep 4911 ax-nul 4919 ax-pow 4970 ax-pr 5033 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1070 df-3an 1071 df-tru 1632 df-ex 1851 df-nf 1856 df-sb 2048 df-eu 2620 df-mo 2621 df-clab 2756 df-cleq 2762 df-clel 2765 df-nfc 2900 df-ne 2942 df-nel 3045 df-ral 3064 df-rex 3065 df-reu 3066 df-rab 3068 df-v 3350 df-sbc 3585 df-csb 3680 df-dif 3723 df-un 3725 df-in 3727 df-ss 3734 df-pss 3736 df-nul 4061 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-ot 4322 df-uni 4572 df-int 4609 df-iun 4653 df-iin 4654 df-br 4784 df-opab 4844 df-mpt 4861 df-tr 4884 df-id 5156 df-eprel 5161 df-po 5169 df-so 5170 df-fr 5207 df-we 5209 df-xp 5254 df-rel 5255 df-cnv 5256 df-co 5257 df-dm 5258 df-rn 5259 df-res 5260 df-ima 5261 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-er 7894 df-ec 7896 df-map 8009 df-pm 8010 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-card 8963 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-n0 11493 df-z 11578 df-uz 11888 df-fz 12533 df-fzo 12673 df-hash 13325 df-word 13498 df-concat 13500 df-s1 13501 df-substr 13502 df-splice 13503 df-s2 13805 df-efg 18335

This theorem is referenced by: frgpcpbl 18385

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