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Theorem efgred2 19795

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 19781	. . . . . . 7 ⊢ 𝑆:dom 𝑆–onto→𝑊
8		fof 6834	. . . . . . 7 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆:dom 𝑆⟶𝑊)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 𝑆:dom 𝑆⟶𝑊
10	9	ffvelcdmi 7117	. . . . 5 ⊢ (𝐵 ∈ dom 𝑆 → (𝑆‘𝐵) ∈ 𝑊)
11	10	ad2antlr 726	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝑆‘𝐵) ∈ 𝑊)
12	1, 2, 3, 4, 5, 6	efgredeu 19794	. . . 4 ⊢ ((𝑆‘𝐵) ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝐵))
13		reurmo 3391	. . . 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 19772	. . . . 5 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1)))))
16	15	simp2bi 1146	. . . 4 ⊢ (𝐴 ∈ dom 𝑆 → (𝐴‘0) ∈ 𝐷)
17	16	ad2antrr 725	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∈ 𝐷)
18	1, 2	efger 19760	. . . . 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 19776	. . . . 5 ⊢ (𝐴 ∈ dom 𝑆 → (𝐴‘0) ∼ (𝑆‘𝐴))
21	20	ad2antrr 725	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∼ (𝑆‘𝐴))
22		simpr 484	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝑆‘𝐴) ∼ (𝑆‘𝐵))
23	19, 21, 22	ertrd 8779	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∼ (𝑆‘𝐵))
24	1, 2, 3, 4, 5, 6	efgsdm 19772	. . . . 5 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1)))))
25	24	simp2bi 1146	. . . 4 ⊢ (𝐵 ∈ dom 𝑆 → (𝐵‘0) ∈ 𝐷)
26	25	ad2antlr 726	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐵‘0) ∈ 𝐷)
27	1, 2, 3, 4, 5, 6	efgsrel 19776	. . . 4 ⊢ (𝐵 ∈ dom 𝑆 → (𝐵‘0) ∼ (𝑆‘𝐵))
28	27	ad2antlr 726	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐵‘0) ∼ (𝑆‘𝐵))
29		breq1 5169	. . . 4 ⊢ (𝑑 = (𝐴‘0) → (𝑑 ∼ (𝑆‘𝐵) ↔ (𝐴‘0) ∼ (𝑆‘𝐵)))
30		breq1 5169	. . . 4 ⊢ (𝑑 = (𝐵‘0) → (𝑑 ∼ (𝑆‘𝐵) ↔ (𝐵‘0) ∼ (𝑆‘𝐵)))
31	29, 30	rmoi 3913	. . 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 1379	. 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 725	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴‘0) ∼ (𝑆‘𝐴))
35		simpr 484	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴‘0) = (𝐵‘0))
36	27	ad2antlr 726	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐵‘0) ∼ (𝑆‘𝐵))
37	35, 36	eqbrtrd 5188	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴‘0) ∼ (𝑆‘𝐵))
38	33, 34, 37	ertr3d 8781	. 2 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝑆‘𝐴) ∼ (𝑆‘𝐵))
39	32, 38	impbida 800	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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃!wreu 3386 ∃*wrmo 3387 {crab 3443 ∖ cdif 3973 ∅c0 4352 {csn 4648 ⟨cop 4654 ⟨cotp 4656 ∪ ciun 5015 class class class wbr 5166 ↦ cmpt 5249 I cid 5592 × cxp 5698 dom cdm 5700 ran crn 5701 ⟶wf 6569 –onto→wfo 6571 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 1_oc1o 8515 2_oc2o 8516 Er wer 8760 0cc0 11184 1c1 11185 − cmin 11520 ...cfz 13567 ..^cfzo 13711 ♯chash 14379 Word cword 14562 splice csplice 14797 ⟨“cs2 14890 ~_FG cefg 19748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-ec 8765 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-concat 14619 df-s1 14644 df-substr 14689 df-pfx 14719 df-splice 14798 df-s2 14897 df-efg 19751

This theorem is referenced by: (None)

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