efgcpbl2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > efgcpbl2		Structured version Visualization version GIF version

Theorem efgcpbl2 19732

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 19693	. . 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 8655	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ 𝑊)
7		wrd0 14501	. . . . 5 ⊢ ∅ ∈ Word (𝐼 × 2_o)
8	1	efgrcl 19690	. . . . . . 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 2843	. . . 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 19731	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ ∅ ∈ 𝑊 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
18	6, 11, 12, 17	syl3anc 1374	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((𝐴 ++ 𝐵) ++ ∅) ∼ ((𝐴 ++ 𝑌) ++ ∅))
19	6, 10	eleqtrd 2838	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐴 ∈ Word (𝐼 × 2_o))
20	4, 12	ercl 8655	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ 𝑊)
21	20, 10	eleqtrd 2838	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝐵 ∈ Word (𝐼 × 2_o))
22		ccatcl 14536	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_o) ∧ 𝐵 ∈ Word (𝐼 × 2_o)) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o))
23	19, 21, 22	syl2anc 585	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∈ Word (𝐼 × 2_o))
24		ccatrid 14550	. . . 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 8657	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ 𝑊)
27	26, 10	eleqtrd 2838	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑌 ∈ Word (𝐼 × 2_o))
28		ccatcl 14536	. . . . 5 ⊢ ((𝐴 ∈ Word (𝐼 × 2_o) ∧ 𝑌 ∈ Word (𝐼 × 2_o)) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o))
29	19, 27, 28	syl2anc 585	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∈ Word (𝐼 × 2_o))
30		ccatrid 14550	. . . 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 5116	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝐴 ++ 𝑌))
33	1, 2, 13, 14, 15, 16	efgcpbl 19731	. . . 4 ⊢ ((∅ ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ 𝐴 ∼ 𝑋) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
34	11, 26, 5, 33	syl3anc 1374	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) ∼ ((∅ ++ 𝑋) ++ 𝑌))
35		ccatlid 14549	. . . . 5 ⊢ (𝐴 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝐴) = 𝐴)
36	19, 35	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝐴) = 𝐴)
37	36	oveq1d 7382	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝐴) ++ 𝑌) = (𝐴 ++ 𝑌))
38	4, 5	ercl2 8657	. . . . . 6 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ 𝑊)
39	38, 10	eleqtrd 2838	. . . . 5 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → 𝑋 ∈ Word (𝐼 × 2_o))
40		ccatlid 14549	. . . . 5 ⊢ (𝑋 ∈ Word (𝐼 × 2_o) → (∅ ++ 𝑋) = 𝑋)
41	39, 40	syl 17	. . . 4 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (∅ ++ 𝑋) = 𝑋)
42	41	oveq1d 7382	. . 3 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → ((∅ ++ 𝑋) ++ 𝑌) = (𝑋 ++ 𝑌))
43	34, 37, 42	3brtr3d 5116	. 2 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝑌) ∼ (𝑋 ++ 𝑌))
44	4, 32, 43	ertrd 8660	1 ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝑋 ++ 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 {crab 3389 Vcvv 3429 ∖ cdif 3886 ∅c0 4273 {csn 4567 ⟨cop 4573 ⟨cotp 4575 ∪ ciun 4933 class class class wbr 5085 ↦ cmpt 5166 I cid 5525 × cxp 5629 ran crn 5632 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 1_oc1o 8398 2_oc2o 8399 Er wer 8640 0cc0 11038 1c1 11039 − cmin 11377 ...cfz 13461 ..^cfzo 13608 ♯chash 14292 Word cword 14475 ++ cconcat 14532 splice csplice 14711 ⟨“cs2 14803 ~_FG cefg 19681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-ec 8645 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476 df-concat 14533 df-s1 14559 df-substr 14604 df-pfx 14634 df-splice 14712 df-s2 14810 df-efg 19684

This theorem is referenced by: frgpcpbl 19734

W3C validator