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Mirrors > Home > MPE Home > Th. List > reust | Structured version Visualization version Unicode version |
Description: The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
Ref | Expression |
---|---|
reust |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-refld 19183 |
. . . 4
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2 | 1 | fveq2i 5850 |
. . 3
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3 | reex 9616 |
. . . 4
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4 | ressuss 21288 |
. . . 4
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5 | 3, 4 | ax-mp 5 |
. . 3
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6 | eqid 2451 |
. . . . 5
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7 | 6 | cnflduss 22333 |
. . . 4
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8 | 7 | oveq1i 6285 |
. . 3
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9 | 2, 5, 8 | 3eqtri 2477 |
. 2
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10 | 0re 9629 |
. . . 4
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11 | 10 | ne0ii 3705 |
. . 3
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12 | cnxmet 21803 |
. . . 4
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13 | xmetpsmet 21373 |
. . . 4
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14 | 12, 13 | ax-mp 5 |
. . 3
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15 | ax-resscn 9582 |
. . 3
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16 | restmetu 21595 |
. . 3
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17 | 11, 14, 15, 16 | mp3an 1368 |
. 2
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18 | reds 19194 |
. . . 4
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19 | 18 | reseq1i 5078 |
. . 3
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20 | 19 | fveq2i 5850 |
. 2
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21 | 9, 17, 20 | 3eqtri 2477 |
1
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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 1672 ax-4 1685 ax-5 1761 ax-6 1808 ax-7 1854 ax-8 1892 ax-9 1899 ax-10 1918 ax-11 1923 ax-12 1936 ax-13 2091 ax-ext 2431 ax-rep 4486 ax-sep 4496 ax-nul 4505 ax-pow 4553 ax-pr 4611 ax-un 6570 ax-cnex 9581 ax-resscn 9582 ax-1cn 9583 ax-icn 9584 ax-addcl 9585 ax-addrcl 9586 ax-mulcl 9587 ax-mulrcl 9588 ax-mulcom 9589 ax-addass 9590 ax-mulass 9591 ax-distr 9592 ax-i2m1 9593 ax-1ne0 9594 ax-1rid 9595 ax-rnegex 9596 ax-rrecex 9597 ax-cnre 9598 ax-pre-lttri 9599 ax-pre-lttrn 9600 ax-pre-ltadd 9601 ax-pre-mulgt0 9602 ax-pre-sup 9603 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 987 df-3an 988 df-tru 1450 df-ex 1667 df-nf 1671 df-sb 1801 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3014 df-sbc 3235 df-csb 3331 df-dif 3374 df-un 3376 df-in 3378 df-ss 3385 df-pss 3387 df-nul 3699 df-if 3849 df-pw 3920 df-sn 3936 df-pr 3938 df-tp 3940 df-op 3942 df-uni 4168 df-int 4204 df-iun 4249 df-br 4374 df-opab 4433 df-mpt 4434 df-tr 4469 df-eprel 4722 df-id 4726 df-po 4732 df-so 4733 df-fr 4770 df-we 4772 df-xp 4817 df-rel 4818 df-cnv 4819 df-co 4820 df-dm 4821 df-rn 4822 df-res 4823 df-ima 4824 df-pred 5358 df-ord 5404 df-on 5405 df-lim 5406 df-suc 5407 df-iota 5524 df-fun 5562 df-fn 5563 df-f 5564 df-f1 5565 df-fo 5566 df-f1o 5567 df-fv 5568 df-riota 6237 df-ov 6278 df-oprab 6279 df-mpt2 6280 df-om 6680 df-1st 6780 df-2nd 6781 df-wrecs 7014 df-recs 7076 df-rdg 7114 df-1o 7168 df-oadd 7172 df-er 7349 df-map 7460 df-en 7556 df-dom 7557 df-sdom 7558 df-fin 7559 df-sup 7942 df-pnf 9663 df-mnf 9664 df-xr 9665 df-ltxr 9666 df-le 9667 df-sub 9848 df-neg 9849 df-div 10258 df-nn 10598 df-2 10656 df-3 10657 df-4 10658 df-5 10659 df-6 10660 df-7 10661 df-8 10662 df-9 10663 df-10 10664 df-n0 10859 df-z 10927 df-dec 11041 df-uz 11149 df-rp 11292 df-xneg 11398 df-xadd 11399 df-xmul 11400 df-ico 11630 df-fz 11775 df-seq 12207 df-exp 12266 df-cj 13172 df-re 13173 df-im 13174 df-sqrt 13308 df-abs 13309 df-struct 15133 df-ndx 15134 df-slot 15135 df-base 15136 df-sets 15137 df-ress 15138 df-plusg 15213 df-mulr 15214 df-starv 15215 df-tset 15219 df-ple 15220 df-ds 15222 df-unif 15223 df-rest 15331 df-psmet 18972 df-xmet 18973 df-met 18974 df-fbas 18977 df-fg 18978 df-metu 18979 df-cnfld 18981 df-refld 19183 df-fil 20871 df-ust 21225 df-uss 21281 |
This theorem is referenced by: recusp 22351 rerrext 28819 |
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