![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rpnnen1lem4 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen1 12972. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rpnnen1lem.1 | β’ π = {π β β€ β£ (π / π) < π₯} |
rpnnen1lem.2 | β’ πΉ = (π₯ β β β¦ (π β β β¦ (sup(π, β, < ) / π))) |
rpnnen1lem.n | β’ β β V |
rpnnen1lem.q | β’ β β V |
Ref | Expression |
---|---|
rpnnen1lem4 | β’ (π₯ β β β sup(ran (πΉβπ₯), β, < ) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpnnen1lem.1 | . . . . 5 β’ π = {π β β€ β£ (π / π) < π₯} | |
2 | rpnnen1lem.2 | . . . . 5 β’ πΉ = (π₯ β β β¦ (π β β β¦ (sup(π, β, < ) / π))) | |
3 | rpnnen1lem.n | . . . . 5 β’ β β V | |
4 | rpnnen1lem.q | . . . . 5 β’ β β V | |
5 | 1, 2, 3, 4 | rpnnen1lem1 12967 | . . . 4 β’ (π₯ β β β (πΉβπ₯) β (β βm β)) |
6 | 4, 3 | elmap 8868 | . . . 4 β’ ((πΉβπ₯) β (β βm β) β (πΉβπ₯):ββΆβ) |
7 | 5, 6 | sylib 217 | . . 3 β’ (π₯ β β β (πΉβπ₯):ββΆβ) |
8 | frn 6725 | . . . 4 β’ ((πΉβπ₯):ββΆβ β ran (πΉβπ₯) β β) | |
9 | qssre 12948 | . . . 4 β’ β β β | |
10 | 8, 9 | sstrdi 3995 | . . 3 β’ ((πΉβπ₯):ββΆβ β ran (πΉβπ₯) β β) |
11 | 7, 10 | syl 17 | . 2 β’ (π₯ β β β ran (πΉβπ₯) β β) |
12 | 1nn 12228 | . . . . . 6 β’ 1 β β | |
13 | 12 | ne0ii 4338 | . . . . 5 β’ β β β |
14 | fdm 6727 | . . . . . 6 β’ ((πΉβπ₯):ββΆβ β dom (πΉβπ₯) = β) | |
15 | 14 | neeq1d 2999 | . . . . 5 β’ ((πΉβπ₯):ββΆβ β (dom (πΉβπ₯) β β β β β β )) |
16 | 13, 15 | mpbiri 257 | . . . 4 β’ ((πΉβπ₯):ββΆβ β dom (πΉβπ₯) β β ) |
17 | dm0rn0 5925 | . . . . 5 β’ (dom (πΉβπ₯) = β β ran (πΉβπ₯) = β ) | |
18 | 17 | necon3bii 2992 | . . . 4 β’ (dom (πΉβπ₯) β β β ran (πΉβπ₯) β β ) |
19 | 16, 18 | sylib 217 | . . 3 β’ ((πΉβπ₯):ββΆβ β ran (πΉβπ₯) β β ) |
20 | 7, 19 | syl 17 | . 2 β’ (π₯ β β β ran (πΉβπ₯) β β ) |
21 | 1, 2, 3, 4 | rpnnen1lem3 12968 | . . 3 β’ (π₯ β β β βπ β ran (πΉβπ₯)π β€ π₯) |
22 | breq2 5153 | . . . . 5 β’ (π¦ = π₯ β (π β€ π¦ β π β€ π₯)) | |
23 | 22 | ralbidv 3176 | . . . 4 β’ (π¦ = π₯ β (βπ β ran (πΉβπ₯)π β€ π¦ β βπ β ran (πΉβπ₯)π β€ π₯)) |
24 | 23 | rspcev 3613 | . . 3 β’ ((π₯ β β β§ βπ β ran (πΉβπ₯)π β€ π₯) β βπ¦ β β βπ β ran (πΉβπ₯)π β€ π¦) |
25 | 21, 24 | mpdan 684 | . 2 β’ (π₯ β β β βπ¦ β β βπ β ran (πΉβπ₯)π β€ π¦) |
26 | suprcl 12179 | . 2 β’ ((ran (πΉβπ₯) β β β§ ran (πΉβπ₯) β β β§ βπ¦ β β βπ β ran (πΉβπ₯)π β€ π¦) β sup(ran (πΉβπ₯), β, < ) β β) | |
27 | 11, 20, 25, 26 | syl3anc 1370 | 1 β’ (π₯ β β β sup(ran (πΉβπ₯), β, < ) β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β wne 2939 βwral 3060 βwrex 3069 {crab 3431 Vcvv 3473 β wss 3949 β c0 4323 class class class wbr 5149 β¦ cmpt 5232 dom cdm 5677 ran crn 5678 βΆwf 6540 βcfv 6544 (class class class)co 7412 βm cmap 8823 supcsup 9438 βcr 11112 1c1 11114 < clt 11253 β€ cle 11254 / cdiv 11876 βcn 12217 β€cz 12563 βcq 12937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9440 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-n0 12478 df-z 12564 df-q 12938 |
This theorem is referenced by: rpnnen1lem5 12970 |
Copyright terms: Public domain | W3C validator |