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Mirrors > Home > MPE Home > Th. List > refld | Structured version Visualization version Unicode version |
Description: The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
Ref | Expression |
---|---|
refld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubdrg 19169 |
. . 3
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2 | 1 | simpri 464 |
. 2
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3 | df-refld 19166 |
. . 3
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4 | cncrng 18982 |
. . . 4
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5 | 1 | simpli 460 |
. . . 4
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6 | eqid 2450 |
. . . . 5
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7 | 6 | subrgcrng 18005 |
. . . 4
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8 | 4, 5, 7 | mp2an 677 |
. . 3
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9 | 3, 8 | eqeltri 2524 |
. 2
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10 | isfld 17977 |
. 2
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11 | 2, 9, 10 | mpbir2an 930 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-cnex 9592 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 ax-addf 9615 ax-mulf 9616 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-int 4234 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-1st 6790 df-2nd 6791 df-tpos 6970 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-1o 7179 df-oadd 7183 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-fin 7570 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-div 10267 df-nn 10607 df-2 10665 df-3 10666 df-4 10667 df-5 10668 df-6 10669 df-7 10670 df-8 10671 df-9 10672 df-10 10673 df-n0 10867 df-z 10935 df-dec 11049 df-uz 11157 df-fz 11782 df-struct 15116 df-ndx 15117 df-slot 15118 df-base 15119 df-sets 15120 df-ress 15121 df-plusg 15196 df-mulr 15197 df-starv 15198 df-tset 15202 df-ple 15203 df-ds 15205 df-unif 15206 df-0g 15333 df-mgm 16481 df-sgrp 16520 df-mnd 16530 df-grp 16666 df-minusg 16667 df-subg 16807 df-cmn 17425 df-mgp 17717 df-ur 17729 df-ring 17775 df-cring 17776 df-oppr 17844 df-dvdsr 17862 df-unit 17863 df-invr 17893 df-dvr 17904 df-drng 17970 df-field 17971 df-subrg 17999 df-cnfld 18964 df-refld 19166 |
This theorem is referenced by: recrng 19182 rrxbase 22340 rrxprds 22341 rrxip 22342 rrxcph 22344 reofld 28596 rearchi 28598 |
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