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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwnconn1lem2 | Structured version Visualization version Unicode version |
Description: Lemma for btwnconn1 32208. Now, we show that two of the hypotheticals we introduced in the first lemma are identical. (Contributed by Scott Fenton, 8-Oct-2013.) |
Ref | Expression |
---|---|
btwnconn1lem2 | Cgr Cgr Cgr Cgr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1ll 1124 | . . 3 Cgr Cgr Cgr Cgr | |
2 | 1 | adantl 482 | . 2 Cgr Cgr Cgr Cgr |
3 | simp11 1091 | . . . 4 | |
4 | simp12 1092 | . . . 4 | |
5 | simp13 1093 | . . . 4 | |
6 | simp21 1094 | . . . 4 | |
7 | simp33 1099 | . . . 4 | |
8 | simp1rl 1126 | . . . . 5 Cgr Cgr Cgr Cgr | |
9 | 8 | adantl 482 | . . . 4 Cgr Cgr Cgr Cgr |
10 | simp2rl 1130 | . . . . . . 7 Cgr Cgr Cgr Cgr | |
11 | simp3rl 1134 | . . . . . . 7 Cgr Cgr Cgr Cgr | |
12 | 10, 11 | jca 554 | . . . . . 6 Cgr Cgr Cgr Cgr |
13 | 12 | adantl 482 | . . . . 5 Cgr Cgr Cgr Cgr |
14 | simp31 1097 | . . . . . . 7 | |
15 | btwnexch 32132 | . . . . . . 7 | |
16 | 3, 4, 6, 14, 7, 15 | syl122anc 1335 | . . . . . 6 |
17 | 16 | adantr 481 | . . . . 5 Cgr Cgr Cgr Cgr |
18 | 13, 17 | mpd 15 | . . . 4 Cgr Cgr Cgr Cgr |
19 | 3, 4, 5, 6, 7, 9, 18 | btwnexchand 32133 | . . 3 Cgr Cgr Cgr Cgr |
20 | 3, 5, 7 | cgrrflx2d 32091 | . . . 4 Cgr |
21 | 20 | adantr 481 | . . 3 Cgr Cgr Cgr Cgr Cgr |
22 | 19, 21 | jca 554 | . 2 Cgr Cgr Cgr Cgr Cgr |
23 | simp23 1096 | . . . 4 | |
24 | simp32 1098 | . . . 4 | |
25 | simp1rr 1127 | . . . . . . 7 Cgr Cgr Cgr Cgr | |
26 | simp2ll 1128 | . . . . . . 7 Cgr Cgr Cgr Cgr | |
27 | 25, 26 | jca 554 | . . . . . 6 Cgr Cgr Cgr Cgr |
28 | 27 | adantl 482 | . . . . 5 Cgr Cgr Cgr Cgr |
29 | simp22 1095 | . . . . . . 7 | |
30 | btwnexch 32132 | . . . . . . 7 | |
31 | 3, 4, 5, 29, 23, 30 | syl122anc 1335 | . . . . . 6 |
32 | 31 | adantr 481 | . . . . 5 Cgr Cgr Cgr Cgr |
33 | 28, 32 | mpd 15 | . . . 4 Cgr Cgr Cgr Cgr |
34 | simp3ll 1132 | . . . . 5 Cgr Cgr Cgr Cgr | |
35 | 34 | adantl 482 | . . . 4 Cgr Cgr Cgr Cgr |
36 | 3, 4, 5, 23, 24, 33, 35 | btwnexchand 32133 | . . 3 Cgr Cgr Cgr Cgr |
37 | 3, 4, 5, 23, 24, 33, 35 | btwnexch3and 32128 | . . . 4 Cgr Cgr Cgr Cgr |
38 | 3, 4, 5, 6, 7, 9, 18 | btwnexch3and 32128 | . . . . 5 Cgr Cgr Cgr Cgr |
39 | 3, 6, 5, 7, 38 | btwncomand 32122 | . . . 4 Cgr Cgr Cgr Cgr |
40 | btwnconn1lem1 32194 | . . . 4 Cgr Cgr Cgr Cgr Cgr | |
41 | simp3lr 1133 | . . . . 5 Cgr Cgr Cgr Cgr Cgr | |
42 | 41 | adantl 482 | . . . 4 Cgr Cgr Cgr Cgr Cgr |
43 | 3, 5, 23, 24, 7, 6, 5, 37, 39, 40, 42 | cgrextendand 32116 | . . 3 Cgr Cgr Cgr Cgr Cgr |
44 | 36, 43 | jca 554 | . 2 Cgr Cgr Cgr Cgr Cgr |
45 | segconeq 32117 | . . . 4 Cgr Cgr | |
46 | 3, 5, 7, 5, 4, 7, 24, 45 | syl133anc 1349 | . . 3 Cgr Cgr |
47 | 46 | adantr 481 | . 2 Cgr Cgr Cgr Cgr Cgr Cgr |
48 | 2, 22, 44, 47 | mp3and 1427 | 1 Cgr Cgr Cgr Cgr |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 cop 4183 class class class wbr 4653 cfv 5888 cn 11020 cee 25768 cbtwn 25769 Cgrccgr 25770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-ee 25771 df-btwn 25772 df-cgr 25773 df-ofs 32090 |
This theorem is referenced by: btwnconn1lem14 32207 |
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