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Theorem efgred2 19720

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
efgred2	⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → ((𝑆‘𝐴) ∼ (𝑆‘𝐵) ↔ (𝐴‘0) = (𝐵‘0)))

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)

Proof of Theorem efgred2 Dummy variables 𝑑 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
2		efgval.r	. . . . . . . 8 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
4		efgval2.t	. . . . . . . 8 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5		efgred.d	. . . . . . . 8 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
6		efgred.s	. . . . . . . 8 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
7	1, 2, 3, 4, 5, 6	efgsfo 19706	. . . . . . 7 ⊢ 𝑆:dom 𝑆–onto→𝑊
8		fof 6740	. . . . . . 7 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆:dom 𝑆⟶𝑊)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 𝑆:dom 𝑆⟶𝑊
10	9	ffvelcdmi 7025	. . . . 5 ⊢ (𝐵 ∈ dom 𝑆 → (𝑆‘𝐵) ∈ 𝑊)
11	10	ad2antlr 733	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝑆‘𝐵) ∈ 𝑊)
12	1, 2, 3, 4, 5, 6	efgredeu 19719	. . . 4 ⊢ ((𝑆‘𝐵) ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝐵))
13		reurmo 3347	. . . 4 ⊢ (∃!𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝐵) → ∃*𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝐵))
14	11, 12, 13	3syl 18	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → ∃*𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝐵))
15	1, 2, 3, 4, 5, 6	efgsdm 19697	. . . . 5 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1)))))
16	15	simp2bi 1152	. . . 4 ⊢ (𝐴 ∈ dom 𝑆 → (𝐴‘0) ∈ 𝐷)
17	16	ad2antrr 732	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∈ 𝐷)
18	1, 2	efger 19685	. . . . 5 ⊢ ∼ Er 𝑊
19	18	a1i 11	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → ∼ Er 𝑊)
20	1, 2, 3, 4, 5, 6	efgsrel 19701	. . . . 5 ⊢ (𝐴 ∈ dom 𝑆 → (𝐴‘0) ∼ (𝑆‘𝐴))
21	20	ad2antrr 732	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∼ (𝑆‘𝐴))
22		simpr 485	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝑆‘𝐴) ∼ (𝑆‘𝐵))
23	19, 21, 22	ertrd 8651	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∼ (𝑆‘𝐵))
24	1, 2, 3, 4, 5, 6	efgsdm 19697	. . . . 5 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1)))))
25	24	simp2bi 1152	. . . 4 ⊢ (𝐵 ∈ dom 𝑆 → (𝐵‘0) ∈ 𝐷)
26	25	ad2antlr 733	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐵‘0) ∈ 𝐷)
27	1, 2, 3, 4, 5, 6	efgsrel 19701	. . . 4 ⊢ (𝐵 ∈ dom 𝑆 → (𝐵‘0) ∼ (𝑆‘𝐵))
28	27	ad2antlr 733	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐵‘0) ∼ (𝑆‘𝐵))
29		breq1 5076	. . . 4 ⊢ (𝑑 = (𝐴‘0) → (𝑑 ∼ (𝑆‘𝐵) ↔ (𝐴‘0) ∼ (𝑆‘𝐵)))
30		breq1 5076	. . . 4 ⊢ (𝑑 = (𝐵‘0) → (𝑑 ∼ (𝑆‘𝐵) ↔ (𝐵‘0) ∼ (𝑆‘𝐵)))
31	29, 30	rmoi 3823	. . 3 ⊢ ((∃*𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝐵) ∧ ((𝐴‘0) ∈ 𝐷 ∧ (𝐴‘0) ∼ (𝑆‘𝐵)) ∧ ((𝐵‘0) ∈ 𝐷 ∧ (𝐵‘0) ∼ (𝑆‘𝐵))) → (𝐴‘0) = (𝐵‘0))
32	14, 17, 23, 26, 28, 31	syl122anc 1387	. 2 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) = (𝐵‘0))
33	18	a1i 11	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → ∼ Er 𝑊)
34	20	ad2antrr 732	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴‘0) ∼ (𝑆‘𝐴))
35		simpr 485	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴‘0) = (𝐵‘0))
36	27	ad2antlr 733	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐵‘0) ∼ (𝑆‘𝐵))
37	35, 36	eqbrtrd 5095	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴‘0) ∼ (𝑆‘𝐵))
38	33, 34, 37	ertr3d 8653	. 2 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝑆‘𝐴) ∼ (𝑆‘𝐵))
39	32, 38	impbida 806	1 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → ((𝑆‘𝐴) ∼ (𝑆‘𝐵) ↔ (𝐴‘0) = (𝐵‘0)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃!wreu 3342 ∃*wrmo 3343 {crab 3391 ∖ cdif 3880 ∅c0 4262 {csn 4556 ⟨cop 4562 ⟨cotp 4564 ∪ ciun 4922 class class class wbr 5073 ↦ cmpt 5154 I cid 5513 × cxp 5617 dom cdm 5619 ran crn 5620 ⟶wf 6482 –onto→wfo 6484 ‘cfv 6486 (class class class)co 7357 ∈ cmpo 7359 1_oc1o 8389 2_oc2o 8390 Er wer 8631 0cc0 11030 1c1 11031 − cmin 11369 ...cfz 13453 ..^cfzo 13600 ♯chash 14284 Word cword 14467 splice csplice 14703 ⟨“cs2 14795 ~_FG cefg 19673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-ot 4565 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-1st 7932 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-ec 8636 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-2 12236 df-n0 12430 df-xnn0 12503 df-z 12517 df-uz 12781 df-rp 12935 df-fz 13454 df-fzo 13601 df-hash 14285 df-word 14468 df-concat 14525 df-s1 14551 df-substr 14596 df-pfx 14626 df-splice 14704 df-s2 14802 df-efg 19676

This theorem is referenced by: (None)

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