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Mirrors > Home > MPE Home > Th. List > dfac14lem | Structured version Visualization version Unicode version |
Description: Lemma for dfac14 21421. By equipping for some with the particular point topology, we can show that is in the closure of ; hence the sequence is in the product of the closures, and we can utilize this instance of ptcls 21419 to extract an element of the closure of . (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
dfac14lem.i | |
dfac14lem.s | |
dfac14lem.0 | |
dfac14lem.p | |
dfac14lem.r | |
dfac14lem.j | |
dfac14lem.c |
Ref | Expression |
---|---|
dfac14lem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2690 | . . . . . . . . . . 11 | |
2 | eqeq1 2626 | . . . . . . . . . . 11 | |
3 | 1, 2 | imbi12d 334 | . . . . . . . . . 10 |
4 | dfac14lem.r | . . . . . . . . . 10 | |
5 | 3, 4 | elrab2 3366 | . . . . . . . . 9 |
6 | dfac14lem.0 | . . . . . . . . . . . . 13 | |
7 | 6 | adantr 481 | . . . . . . . . . . . 12 |
8 | ineq1 3807 | . . . . . . . . . . . . . 14 | |
9 | ssun1 3776 | . . . . . . . . . . . . . . 15 | |
10 | sseqin2 3817 | . . . . . . . . . . . . . . 15 | |
11 | 9, 10 | mpbi 220 | . . . . . . . . . . . . . 14 |
12 | 8, 11 | syl6eq 2672 | . . . . . . . . . . . . 13 |
13 | 12 | neeq1d 2853 | . . . . . . . . . . . 12 |
14 | 7, 13 | syl5ibrcom 237 | . . . . . . . . . . 11 |
15 | 14 | imim2d 57 | . . . . . . . . . 10 |
16 | 15 | expimpd 629 | . . . . . . . . 9 |
17 | 5, 16 | syl5bi 232 | . . . . . . . 8 |
18 | 17 | ralrimiv 2965 | . . . . . . 7 |
19 | dfac14lem.s | . . . . . . . . . . . 12 | |
20 | snex 4908 | . . . . . . . . . . . 12 | |
21 | unexg 6959 | . . . . . . . . . . . 12 | |
22 | 19, 20, 21 | sylancl 694 | . . . . . . . . . . 11 |
23 | ssun2 3777 | . . . . . . . . . . . 12 | |
24 | dfac14lem.p | . . . . . . . . . . . . . 14 | |
25 | uniexg 6955 | . . . . . . . . . . . . . . 15 | |
26 | pwexg 4850 | . . . . . . . . . . . . . . 15 | |
27 | 19, 25, 26 | 3syl 18 | . . . . . . . . . . . . . 14 |
28 | 24, 27 | syl5eqel 2705 | . . . . . . . . . . . . 13 |
29 | snidg 4206 | . . . . . . . . . . . . 13 | |
30 | 28, 29 | syl 17 | . . . . . . . . . . . 12 |
31 | 23, 30 | sseldi 3601 | . . . . . . . . . . 11 |
32 | epttop 20813 | . . . . . . . . . . 11 TopOn | |
33 | 22, 31, 32 | syl2anc 693 | . . . . . . . . . 10 TopOn |
34 | 4, 33 | syl5eqel 2705 | . . . . . . . . 9 TopOn |
35 | topontop 20718 | . . . . . . . . 9 TopOn | |
36 | 34, 35 | syl 17 | . . . . . . . 8 |
37 | toponuni 20719 | . . . . . . . . . 10 TopOn | |
38 | 34, 37 | syl 17 | . . . . . . . . 9 |
39 | 9, 38 | syl5sseq 3653 | . . . . . . . 8 |
40 | 31, 38 | eleqtrd 2703 | . . . . . . . 8 |
41 | eqid 2622 | . . . . . . . . 9 | |
42 | 41 | elcls 20877 | . . . . . . . 8 |
43 | 36, 39, 40, 42 | syl3anc 1326 | . . . . . . 7 |
44 | 18, 43 | mpbird 247 | . . . . . 6 |
45 | 44 | ralrimiva 2966 | . . . . 5 |
46 | dfac14lem.i | . . . . . 6 | |
47 | mptelixpg 7945 | . . . . . 6 | |
48 | 46, 47 | syl 17 | . . . . 5 |
49 | 45, 48 | mpbird 247 | . . . 4 |
50 | ne0i 3921 | . . . 4 | |
51 | 49, 50 | syl 17 | . . 3 |
52 | dfac14lem.c | . . 3 | |
53 | 34 | ralrimiva 2966 | . . . . 5 TopOn |
54 | dfac14lem.j | . . . . . 6 | |
55 | 54 | pttopon 21399 | . . . . 5 TopOn TopOn |
56 | 46, 53, 55 | syl2anc 693 | . . . 4 TopOn |
57 | topontop 20718 | . . . 4 TopOn | |
58 | cls0 20884 | . . . 4 | |
59 | 56, 57, 58 | 3syl 18 | . . 3 |
60 | 51, 52, 59 | 3netr4d 2871 | . 2 |
61 | fveq2 6191 | . . 3 | |
62 | 61 | necon3i 2826 | . 2 |
63 | 60, 62 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 wral 2912 crab 2916 cvv 3200 cun 3572 cin 3573 wss 3574 c0 3915 cpw 4158 csn 4177 cuni 4436 cmpt 4729 cfv 5888 cixp 7908 cpt 16099 ctop 20698 TopOnctopon 20715 ccl 20822 |
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 |
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-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-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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-ixp 7909 df-en 7956 df-fin 7959 df-fi 8317 df-topgen 16104 df-pt 16105 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 |
This theorem is referenced by: dfac14 21421 |
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