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Mirrors > Home > MPE Home > Th. List > tuslem | Structured version Visualization version Unicode version |
Description: Lemma for tusbas 22072, tusunif 22073, and tustopn 22075. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
Ref | Expression |
---|---|
tuslem.k | toUnifSp |
Ref | Expression |
---|---|
tuslem | UnifOn unifTop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseid 15919 | . . . 4 Slot | |
2 | 1re 10039 | . . . . . 6 | |
3 | 1lt9 11229 | . . . . . 6 | |
4 | 2, 3 | ltneii 10150 | . . . . 5 |
5 | basendx 15923 | . . . . . 6 | |
6 | tsetndx 16040 | . . . . . 6 TopSet | |
7 | 5, 6 | neeq12i 2860 | . . . . 5 TopSet |
8 | 4, 7 | mpbir 221 | . . . 4 TopSet |
9 | 1, 8 | setsnid 15915 | . . 3 sSet TopSet unifTop |
10 | ustbas2 22029 | . . . 4 UnifOn | |
11 | uniexg 6955 | . . . . 5 UnifOn | |
12 | dmexg 7097 | . . . . 5 | |
13 | eqid 2622 | . . . . . 6 | |
14 | df-unif 15965 | . . . . . 6 Slot ; | |
15 | 1nn 11031 | . . . . . . 7 | |
16 | 3nn0 11310 | . . . . . . 7 | |
17 | 1nn0 11308 | . . . . . . 7 | |
18 | 1lt10 11681 | . . . . . . 7 ; | |
19 | 15, 16, 17, 18 | declti 11546 | . . . . . 6 ; |
20 | 3nn 11186 | . . . . . . 7 | |
21 | 17, 20 | decnncl 11518 | . . . . . 6 ; |
22 | 13, 14, 19, 21 | 2strbas 15984 | . . . . 5 |
23 | 11, 12, 22 | 3syl 18 | . . . 4 UnifOn |
24 | 10, 23 | eqtrd 2656 | . . 3 UnifOn |
25 | tuslem.k | . . . . 5 toUnifSp | |
26 | tusval 22070 | . . . . 5 UnifOn toUnifSp sSet TopSet unifTop | |
27 | 25, 26 | syl5eq 2668 | . . . 4 UnifOn sSet TopSet unifTop |
28 | 27 | fveq2d 6195 | . . 3 UnifOn sSet TopSet unifTop |
29 | 9, 24, 28 | 3eqtr4a 2682 | . 2 UnifOn |
30 | unifid 16065 | . . . 4 Slot | |
31 | 9re 11107 | . . . . . 6 | |
32 | 9nn0 11316 | . . . . . . 7 | |
33 | 9lt10 11673 | . . . . . . 7 ; | |
34 | 15, 16, 32, 33 | declti 11546 | . . . . . 6 ; |
35 | 31, 34 | gtneii 10149 | . . . . 5 ; |
36 | unifndx 16064 | . . . . . 6 ; | |
37 | 36, 6 | neeq12i 2860 | . . . . 5 TopSet ; |
38 | 35, 37 | mpbir 221 | . . . 4 TopSet |
39 | 30, 38 | setsnid 15915 | . . 3 sSet TopSet unifTop |
40 | 13, 14, 19, 21 | 2strop 15985 | . . 3 UnifOn |
41 | 27 | fveq2d 6195 | . . 3 UnifOn sSet TopSet unifTop |
42 | 39, 40, 41 | 3eqtr4a 2682 | . 2 UnifOn |
43 | 27 | fveq2d 6195 | . . . 4 UnifOn TopSet TopSet sSet TopSet unifTop |
44 | prex 4909 | . . . . 5 | |
45 | fvex 6201 | . . . . 5 unifTop | |
46 | tsetid 16041 | . . . . . 6 TopSet Slot TopSet | |
47 | 46 | setsid 15914 | . . . . 5 unifTop unifTop TopSet sSet TopSet unifTop |
48 | 44, 45, 47 | mp2an 708 | . . . 4 unifTop TopSet sSet TopSet unifTop |
49 | 43, 48 | syl6reqr 2675 | . . 3 UnifOn unifTop TopSet |
50 | utopbas 22039 | . . . . . 6 UnifOn unifTop | |
51 | 49 | unieqd 4446 | . . . . . 6 UnifOn unifTop TopSet |
52 | 50, 29, 51 | 3eqtr3rd 2665 | . . . . 5 UnifOn TopSet |
53 | 52 | oveq2d 6666 | . . . 4 UnifOn TopSet ↾t TopSet TopSet ↾t |
54 | fvex 6201 | . . . . 5 TopSet | |
55 | eqid 2622 | . . . . . 6 TopSet TopSet | |
56 | 55 | restid 16094 | . . . . 5 TopSet TopSet ↾t TopSet TopSet |
57 | 54, 56 | ax-mp 5 | . . . 4 TopSet ↾t TopSet TopSet |
58 | eqid 2622 | . . . . 5 | |
59 | eqid 2622 | . . . . 5 TopSet TopSet | |
60 | 58, 59 | topnval 16095 | . . . 4 TopSet ↾t |
61 | 53, 57, 60 | 3eqtr3g 2679 | . . 3 UnifOn TopSet |
62 | 49, 61 | eqtrd 2656 | . 2 UnifOn unifTop |
63 | 29, 42, 62 | 3jca 1242 | 1 UnifOn unifTop |
Colors of variables: wff setvar class |
Syntax hints: wi 4 w3a 1037 wceq 1483 wcel 1990 wne 2794 cvv 3200 cpr 4179 cop 4183 cuni 4436 cdm 5114 cfv 5888 (class class class)co 6650 c1 9937 c3 11071 c9 11077 ;cdc 11493 cnx 15854 sSet csts 15855 cbs 15857 TopSetcts 15947 cunif 15951 ↾t crest 16081 ctopn 16082 UnifOncust 22003 unifTopcutop 22034 toUnifSpctus 22059 |
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-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-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 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-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-tset 15960 df-unif 15965 df-rest 16083 df-topn 16084 df-ust 22004 df-utop 22035 df-tus 22062 |
This theorem is referenced by: tusbas 22072 tusunif 22073 tustopn 22075 tususp 22076 |
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