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Mirrors > Home > MPE Home > Th. List > krippen | Structured version Visualization version Unicode version |
Description: Krippenlemma (German for crib's lemma) Lemma 7.22 of [Schwabhauser] p. 53. proven by Gupta 1965 as Theorem 3.45. (Contributed by Thierry Arnoux, 12-Aug-2019.) |
Ref | Expression |
---|---|
mirval.p | |
mirval.d | |
mirval.i | Itv |
mirval.l | LineG |
mirval.s | pInvG |
mirval.g | TarskiG |
krippen.m | |
krippen.n | |
krippen.a | |
krippen.b | |
krippen.c | |
krippen.e | |
krippen.f | |
krippen.x | |
krippen.y | |
krippen.1 | |
krippen.2 | |
krippen.3 | |
krippen.4 | |
krippen.5 | |
krippen.6 |
Ref | Expression |
---|---|
krippen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . 3 | |
2 | mirval.d | . . 3 | |
3 | mirval.i | . . 3 Itv | |
4 | mirval.l | . . 3 LineG | |
5 | mirval.s | . . 3 pInvG | |
6 | mirval.g | . . . 4 TarskiG | |
7 | 6 | adantr 481 | . . 3 ≤G TarskiG |
8 | krippen.m | . . 3 | |
9 | krippen.n | . . 3 | |
10 | krippen.a | . . . 4 | |
11 | 10 | adantr 481 | . . 3 ≤G |
12 | krippen.b | . . . 4 | |
13 | 12 | adantr 481 | . . 3 ≤G |
14 | krippen.c | . . . 4 | |
15 | 14 | adantr 481 | . . 3 ≤G |
16 | krippen.e | . . . 4 | |
17 | 16 | adantr 481 | . . 3 ≤G |
18 | krippen.f | . . . 4 | |
19 | 18 | adantr 481 | . . 3 ≤G |
20 | krippen.x | . . . 4 | |
21 | 20 | adantr 481 | . . 3 ≤G |
22 | krippen.y | . . . 4 | |
23 | 22 | adantr 481 | . . 3 ≤G |
24 | krippen.1 | . . . 4 | |
25 | 24 | adantr 481 | . . 3 ≤G |
26 | krippen.2 | . . . 4 | |
27 | 26 | adantr 481 | . . 3 ≤G |
28 | krippen.3 | . . . 4 | |
29 | 28 | adantr 481 | . . 3 ≤G |
30 | krippen.4 | . . . 4 | |
31 | 30 | adantr 481 | . . 3 ≤G |
32 | krippen.5 | . . . 4 | |
33 | 32 | adantr 481 | . . 3 ≤G |
34 | krippen.6 | . . . 4 | |
35 | 34 | adantr 481 | . . 3 ≤G |
36 | eqid 2622 | . . 3 ≤G ≤G | |
37 | simpr 477 | . . 3 ≤G ≤G | |
38 | 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 36, 37 | krippenlem 25585 | . 2 ≤G |
39 | 6 | adantr 481 | . . 3 ≤G TarskiG |
40 | 22 | adantr 481 | . . 3 ≤G |
41 | 14 | adantr 481 | . . 3 ≤G |
42 | 20 | adantr 481 | . . 3 ≤G |
43 | 16 | adantr 481 | . . . 4 ≤G |
44 | 18 | adantr 481 | . . . 4 ≤G |
45 | 10 | adantr 481 | . . . 4 ≤G |
46 | 12 | adantr 481 | . . . 4 ≤G |
47 | 24 | adantr 481 | . . . . 5 ≤G |
48 | 1, 2, 3, 39, 45, 41, 43, 47 | tgbtwncom 25383 | . . . 4 ≤G |
49 | 26 | adantr 481 | . . . . 5 ≤G |
50 | 1, 2, 3, 39, 46, 41, 44, 49 | tgbtwncom 25383 | . . . 4 ≤G |
51 | 30 | adantr 481 | . . . 4 ≤G |
52 | 28 | adantr 481 | . . . 4 ≤G |
53 | 34 | adantr 481 | . . . 4 ≤G |
54 | 32 | adantr 481 | . . . 4 ≤G |
55 | simpr 477 | . . . 4 ≤G ≤G | |
56 | 1, 2, 3, 4, 5, 39, 9, 8, 43, 44, 41, 45, 46, 40, 42, 48, 50, 51, 52, 53, 54, 36, 55 | krippenlem 25585 | . . 3 ≤G |
57 | 1, 2, 3, 39, 40, 41, 42, 56 | tgbtwncom 25383 | . 2 ≤G |
58 | 1, 2, 3, 36, 6, 14, 10, 14, 16 | legtrid 25486 | . 2 ≤G ≤G |
59 | 38, 57, 58 | mpjaodan 827 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 class class class wbr 4653 cfv 5888 (class class class)co 6650 cbs 15857 cds 15950 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 ≤Gcleg 25477 pInvGcmir 25547 |
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-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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-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-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-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-cgrg 25406 df-leg 25478 df-mir 25548 |
This theorem is referenced by: footex 25613 mideulem2 25626 |
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