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Mirrors > Home > MPE Home > Th. List > efgred2 | Structured version Visualization version GIF version |
Description: Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.) |
Ref | Expression |
---|---|
efgval.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
efgval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
efgval2.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) |
efgval2.t | ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (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))) |
Ref | Expression |
---|---|
efgred2 | ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → ((𝑆‘𝐴) ∼ (𝑆‘𝐵) ↔ (𝐴‘0) = (𝐵‘0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgval.w | . . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
2 | efgval.r | . . . . . . . 8 ⊢ ∼ = ( ~FG ‘𝐼) | |
3 | efgval2.m | . . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
4 | efgval2.t | . . . . . . . 8 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (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 19083 | . . . . . . 7 ⊢ 𝑆:dom 𝑆–onto→𝑊 |
8 | fof 6611 | . . . . . . 7 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆:dom 𝑆⟶𝑊) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ 𝑆:dom 𝑆⟶𝑊 |
10 | 9 | ffvelrni 6881 | . . . . 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 19096 | . . . 4 ⊢ ((𝑆‘𝐵) ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝐵)) |
13 | reurmo 3330 | . . . 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 19074 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1))))) |
16 | 15 | simp2bi 1148 | . . . 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 19062 | . . . . 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 19078 | . . . . 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 488 | . . . 4 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝑆‘𝐴) ∼ (𝑆‘𝐵)) | |
23 | 19, 21, 22 | ertrd 8385 | . . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝑆‘𝐴) ∼ (𝑆‘𝐵)) → (𝐴‘0) ∼ (𝑆‘𝐵)) |
24 | 1, 2, 3, 4, 5, 6 | efgsdm 19074 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1))))) |
25 | 24 | simp2bi 1148 | . . . 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 19078 | . . . 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 5042 | . . . 4 ⊢ (𝑑 = (𝐴‘0) → (𝑑 ∼ (𝑆‘𝐵) ↔ (𝐴‘0) ∼ (𝑆‘𝐵))) | |
30 | breq1 5042 | . . . 4 ⊢ (𝑑 = (𝐵‘0) → (𝑑 ∼ (𝑆‘𝐵) ↔ (𝐵‘0) ∼ (𝑆‘𝐵))) | |
31 | 29, 30 | rmoi 3790 | . . 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 1381 | . 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 488 | . . . 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 5061 | . . 3 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴‘0) ∼ (𝑆‘𝐵)) |
38 | 33, 34, 37 | ertr3d 8387 | . 2 ⊢ (((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) ∧ (𝐴‘0) = (𝐵‘0)) → (𝑆‘𝐴) ∼ (𝑆‘𝐵)) |
39 | 32, 38 | impbida 801 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃!wreu 3053 ∃*wrmo 3054 {crab 3055 ∖ cdif 3850 ∅c0 4223 {csn 4527 〈cop 4533 〈cotp 4535 ∪ ciun 4890 class class class wbr 5039 ↦ cmpt 5120 I cid 5439 × cxp 5534 dom cdm 5536 ran crn 5537 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 1oc1o 8173 2oc2o 8174 Er wer 8366 0cc0 10694 1c1 10695 − cmin 11027 ...cfz 13060 ..^cfzo 13203 ♯chash 13861 Word cword 14034 splice csplice 14279 〈“cs2 14371 ~FG cefg 19050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-ot 4536 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-ec 8371 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-rp 12552 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-substr 14171 df-pfx 14201 df-splice 14280 df-s2 14378 df-efg 19053 |
This theorem is referenced by: (None) |
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