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Theorem efgred2 19739

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 19725	. . . . . . 7 ⊢ 𝑆:dom 𝑆–onto→𝑊
8		fof 6800	. . . . . . 7 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆:dom 𝑆⟶𝑊)
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 𝑆:dom 𝑆⟶𝑊
10	9	ffvelcdmi 7083	. . . . 5 ⊢ (𝐵 ∈ dom 𝑆 → (𝑆‘𝐵) ∈ 𝑊)
11	10	ad2antlr 727	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝑆‘𝐵) ∈ 𝑊)
12	1, 2, 3, 4, 5, 6	efgredeu 19738	. . . 4 ⊢ ((𝑆‘𝐵) ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝐵))
13		reurmo 3366	. . . 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 19716	. . . . 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 726	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∈ 𝐷)
18	1, 2	efger 19704	. . . . 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 19720	. . . . 5 ⊢ (𝐴 ∈ dom 𝑆 → (𝐴‘0) ∼ (𝑆‘𝐴))
21	20	ad2antrr 726	. . . 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 8743	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∼ (𝑆‘𝐵))
24	1, 2, 3, 4, 5, 6	efgsdm 19716	. . . . 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 727	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐵‘0) ∈ 𝐷)
27	1, 2, 3, 4, 5, 6	efgsrel 19720	. . . 4 ⊢ (𝐵 ∈ dom 𝑆 → (𝐵‘0) ∼ (𝑆‘𝐵))
28	27	ad2antlr 727	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐵‘0) ∼ (𝑆‘𝐵))
29		breq1 5126	. . . 4 ⊢ (𝑑 = (𝐴‘0) → (𝑑 ∼ (𝑆‘𝐵) ↔ (𝐴‘0) ∼ (𝑆‘𝐵)))
30		breq1 5126	. . . 4 ⊢ (𝑑 = (𝐵‘0) → (𝑑 ∼ (𝑆‘𝐵) ↔ (𝐵‘0) ∼ (𝑆‘𝐵)))
31	29, 30	rmoi 3871	. . 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 1380	. 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 726	. . 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 727	. . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐵‘0) ∼ (𝑆‘𝐵))
37	35, 36	eqbrtrd 5145	. . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴‘0) ∼ (𝑆‘𝐵))
38	33, 34, 37	ertr3d 8745	. 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 1539 ∈ wcel 2107 ∀wral 3050 ∃!wreu 3361 ∃*wrmo 3362 {crab 3419 ∖ cdif 3928 ∅c0 4313 {csn 4606 ⟨cop 4612 ⟨cotp 4614 ∪ ciun 4971 class class class wbr 5123 ↦ cmpt 5205 I cid 5557 × cxp 5663 dom cdm 5665 ran crn 5666 ⟶wf 6537 –onto→wfo 6539 ‘cfv 6541 (class class class)co 7413 ∈ cmpo 7415 1_oc1o 8481 2_oc2o 8482 Er wer 8724 0cc0 11137 1c1 11138 − cmin 11474 ...cfz 13529 ..^cfzo 13676 ♯chash 14351 Word cword 14534 splice csplice 14769 ⟨“cs2 14862 ~_FG cefg 19692

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-ot 4615 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-ec 8729 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-n0 12510 df-xnn0 12583 df-z 12597 df-uz 12861 df-rp 13017 df-fz 13530 df-fzo 13677 df-hash 14352 df-word 14535 df-concat 14591 df-s1 14616 df-substr 14661 df-pfx 14691 df-splice 14770 df-s2 14869 df-efg 19695

This theorem is referenced by: (None)

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