Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > preimaleiinlt | Structured version Visualization version Unicode version |
Description: A preimage of a left-open, right-closed, unbounded below interval, expressed as an indexed intersection of preimages of open, unbound below intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
preimaleiinlt.x | |
preimaleiinlt.b | |
preimaleiinlt.c |
Ref | Expression |
---|---|
preimaleiinlt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimaleiinlt.x | . . . 4 | |
2 | simpllr 799 | . . . . . . . . . 10 | |
3 | preimaleiinlt.b | . . . . . . . . . . . 12 | |
4 | 3 | ad2antrr 762 | . . . . . . . . . . 11 |
5 | preimaleiinlt.c | . . . . . . . . . . . . 13 | |
6 | 5 | ad3antrrr 766 | . . . . . . . . . . . 12 |
7 | 6 | rexrd 10089 | . . . . . . . . . . 11 |
8 | 5 | adantr 481 | . . . . . . . . . . . . . 14 |
9 | nnrecre 11057 | . . . . . . . . . . . . . . 15 | |
10 | 9 | adantl 482 | . . . . . . . . . . . . . 14 |
11 | 8, 10 | readdcld 10069 | . . . . . . . . . . . . 13 |
12 | 11 | ad4ant14 1293 | . . . . . . . . . . . 12 |
13 | 12 | rexrd 10089 | . . . . . . . . . . 11 |
14 | simplr 792 | . . . . . . . . . . 11 | |
15 | nnrp 11842 | . . . . . . . . . . . . . . 15 | |
16 | rpreccl 11857 | . . . . . . . . . . . . . . 15 | |
17 | 15, 16 | syl 17 | . . . . . . . . . . . . . 14 |
18 | 17 | adantl 482 | . . . . . . . . . . . . 13 |
19 | 8, 18 | ltaddrpd 11905 | . . . . . . . . . . . 12 |
20 | 19 | ad4ant14 1293 | . . . . . . . . . . 11 |
21 | 4, 7, 13, 14, 20 | xrlelttrd 11991 | . . . . . . . . . 10 |
22 | 2, 21 | jca 554 | . . . . . . . . 9 |
23 | rabid 3116 | . . . . . . . . 9 | |
24 | 22, 23 | sylibr 224 | . . . . . . . 8 |
25 | 24 | ralrimiva 2966 | . . . . . . 7 |
26 | vex 3203 | . . . . . . . 8 | |
27 | eliin 4525 | . . . . . . . 8 | |
28 | 26, 27 | ax-mp 5 | . . . . . . 7 |
29 | 25, 28 | sylibr 224 | . . . . . 6 |
30 | 29 | ex 450 | . . . . 5 |
31 | 30 | ex 450 | . . . 4 |
32 | 1, 31 | ralrimi 2957 | . . 3 |
33 | nfcv 2764 | . . . . 5 | |
34 | nfrab1 3122 | . . . . 5 | |
35 | 33, 34 | nfiin 4549 | . . . 4 |
36 | 35 | rabssf 39302 | . . 3 |
37 | 32, 36 | sylibr 224 | . 2 |
38 | nnn0 39595 | . . . . 5 | |
39 | 38 | a1i 11 | . . . 4 |
40 | iinrab 4582 | . . . 4 | |
41 | 39, 40 | syl 17 | . . 3 |
42 | 3 | ad2antrr 762 | . . . . . . . . . . . 12 |
43 | 11 | ad4ant13 1292 | . . . . . . . . . . . . 13 |
44 | 43 | rexrd 10089 | . . . . . . . . . . . 12 |
45 | simpr 477 | . . . . . . . . . . . 12 | |
46 | 42, 44, 45 | xrltled 39486 | . . . . . . . . . . 11 |
47 | 46 | ex 450 | . . . . . . . . . 10 |
48 | 47 | ralimdva 2962 | . . . . . . . . 9 |
49 | 48 | imp 445 | . . . . . . . 8 |
50 | nfv 1843 | . . . . . . . . . 10 | |
51 | nfra1 2941 | . . . . . . . . . 10 | |
52 | 50, 51 | nfan 1828 | . . . . . . . . 9 |
53 | 3 | adantr 481 | . . . . . . . . 9 |
54 | 5 | ad2antrr 762 | . . . . . . . . 9 |
55 | 52, 53, 54 | xrralrecnnle 39602 | . . . . . . . 8 |
56 | 49, 55 | mpbird 247 | . . . . . . 7 |
57 | 56 | ex 450 | . . . . . 6 |
58 | 57 | ex 450 | . . . . 5 |
59 | 1, 58 | ralrimi 2957 | . . . 4 |
60 | ss2rab 3678 | . . . 4 | |
61 | 59, 60 | sylibr 224 | . . 3 |
62 | 41, 61 | eqsstrd 3639 | . 2 |
63 | 37, 62 | eqssd 3620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wnf 1708 wcel 1990 wne 2794 wral 2912 crab 2916 cvv 3200 wss 3574 c0 3915 ciin 4521 class class class wbr 4653 (class class class)co 6650 cr 9935 c1 9937 caddc 9939 cxr 10073 clt 10074 cle 10075 cdiv 10684 cn 11020 crp 11832 |
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-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 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-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-iun 4522 df-iin 4523 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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 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-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fl 12593 |
This theorem is referenced by: salpreimaltle 40935 |
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