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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0mulc1cn | Structured version Visualization version Unicode version |
Description: The operation multiplying a nonnegative real numbers by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
Ref | Expression |
---|---|
xrge0mulc1cn.k |
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xrge0mulc1cn.f |
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xrge0mulc1cn.c |
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Ref | Expression |
---|---|
xrge0mulc1cn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0mulc1cn.k |
. . . . . 6
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2 | letopon 20276 |
. . . . . . 7
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3 | iccssxr 11751 |
. . . . . . 7
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4 | resttopon 20232 |
. . . . . . 7
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5 | 2, 3, 4 | mp2an 683 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 1, 5 | eqeltri 2536 |
. . . . 5
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7 | 6 | a1i 11 |
. . . 4
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8 | 0e0iccpnf 11778 |
. . . . 5
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9 | 8 | a1i 11 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | simpl 463 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | oveq2d 6336 |
. . . . . . . 8
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12 | simpr 467 |
. . . . . . . . . 10
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13 | 3, 12 | sseldi 3442 |
. . . . . . . . 9
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14 | xmul01 11587 |
. . . . . . . . 9
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15 | 13, 14 | syl 17 |
. . . . . . . 8
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16 | 11, 15 | eqtrd 2496 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | mpteq2dva 4505 |
. . . . . 6
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18 | xrge0mulc1cn.f |
. . . . . 6
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19 | fconstmpt 4900 |
. . . . . 6
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20 | 17, 18, 19 | 3eqtr4g 2521 |
. . . . 5
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21 | c0ex 9668 |
. . . . . 6
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22 | 21 | fconst2 6150 |
. . . . 5
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23 | 20, 22 | sylibr 217 |
. . . 4
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24 | cnconst 20355 |
. . . 4
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25 | 7, 7, 9, 23, 24 | syl22anc 1277 |
. . 3
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26 | 25 | adantl 472 |
. 2
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27 | eqid 2462 |
. . . . . . . . 9
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28 | oveq1 6327 |
. . . . . . . . . 10
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29 | 28 | cbvmptv 4511 |
. . . . . . . . 9
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30 | id 22 |
. . . . . . . . 9
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31 | 27, 29, 30 | xrmulc1cn 28787 |
. . . . . . . 8
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32 | letopuni 20278 |
. . . . . . . . 9
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33 | 32 | cnrest 20356 |
. . . . . . . 8
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34 | 31, 3, 33 | sylancl 673 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | resmpt 5176 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | 3, 35 | ax-mp 5 |
. . . . . . . 8
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37 | 36, 18 | eqtr4i 2487 |
. . . . . . 7
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38 | 1 | eqcomi 2471 |
. . . . . . . 8
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39 | 38 | oveq1i 6330 |
. . . . . . 7
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40 | 34, 37, 39 | 3eltr3g 2556 |
. . . . . 6
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41 | 2 | a1i 11 |
. . . . . . 7
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42 | simpr 467 |
. . . . . . . . . 10
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43 | ioorp 11746 |
. . . . . . . . . . . 12
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44 | ioossicc 11754 |
. . . . . . . . . . . 12
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45 | 43, 44 | eqsstr3i 3475 |
. . . . . . . . . . 11
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46 | simpl 463 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 45, 46 | sseldi 3442 |
. . . . . . . . . 10
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48 | ge0xmulcl 11782 |
. . . . . . . . . 10
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49 | 42, 47, 48 | syl2anc 671 |
. . . . . . . . 9
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50 | 49, 18 | fmptd 6074 |
. . . . . . . 8
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51 | frn 5762 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
52 | 50, 51 | syl 17 |
. . . . . . 7
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53 | 3 | a1i 11 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | cnrest2 20357 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
55 | 41, 52, 53, 54 | syl3anc 1276 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | 40, 55 | mpbid 215 |
. . . . 5
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57 | 1 | oveq2i 6331 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | 56, 57 | syl6eleqr 2551 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
59 | 58, 43 | eleq2s 2558 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 59 | adantl 472 |
. 2
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61 | xrge0mulc1cn.c |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
62 | 0xr 9718 |
. . . 4
![]() ![]() ![]() ![]() | |
63 | pnfxr 11446 |
. . . 4
![]() ![]() ![]() ![]() | |
64 | 0ltpnf 11458 |
. . . 4
![]() ![]() ![]() ![]() | |
65 | elicoelioo 28412 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
66 | 62, 63, 64, 65 | mp3an 1373 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
67 | 61, 66 | sylib 201 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
68 | 26, 60, 67 | mpjaodan 800 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4531 ax-sep 4541 ax-nul 4550 ax-pow 4598 ax-pr 4656 ax-un 6615 ax-cnex 9626 ax-resscn 9627 ax-1cn 9628 ax-icn 9629 ax-addcl 9630 ax-addrcl 9631 ax-mulcl 9632 ax-mulrcl 9633 ax-mulcom 9634 ax-addass 9635 ax-mulass 9636 ax-distr 9637 ax-i2m1 9638 ax-1ne0 9639 ax-1rid 9640 ax-rnegex 9641 ax-rrecex 9642 ax-cnre 9643 ax-pre-lttri 9644 ax-pre-lttrn 9645 ax-pre-ltadd 9646 ax-pre-mulgt0 9647 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rmo 2757 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-iin 4295 df-br 4419 df-opab 4478 df-mpt 4479 df-tr 4514 df-eprel 4767 df-id 4771 df-po 4777 df-so 4778 df-fr 4815 df-we 4817 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-rn 4867 df-res 4868 df-ima 4869 df-pred 5403 df-ord 5449 df-on 5450 df-lim 5451 df-suc 5452 df-iota 5569 df-fun 5607 df-fn 5608 df-f 5609 df-f1 5610 df-fo 5611 df-f1o 5612 df-fv 5613 df-isom 5614 df-riota 6282 df-ov 6323 df-oprab 6324 df-mpt2 6325 df-om 6725 df-1st 6825 df-2nd 6826 df-wrecs 7059 df-recs 7121 df-rdg 7159 df-1o 7213 df-oadd 7217 df-er 7394 df-map 7505 df-en 7601 df-dom 7602 df-sdom 7603 df-fin 7604 df-fi 7956 df-pnf 9708 df-mnf 9709 df-xr 9710 df-ltxr 9711 df-le 9712 df-sub 9893 df-neg 9894 df-div 10303 df-rp 11337 df-xneg 11443 df-xmul 11445 df-ioo 11673 df-ico 11675 df-icc 11676 df-rest 15376 df-topgen 15397 df-ordt 15454 df-ps 16501 df-tsr 16502 df-top 19976 df-bases 19977 df-topon 19978 df-cn 20298 df-cnp 20299 |
This theorem is referenced by: esummulc1 28953 |
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