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Mirrors > Home > MPE Home > Th. List > rexneg | Structured version Unicode version |
Description: Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
rexneg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 11192 |
. 2
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2 | renepnf 9534 |
. . . 4
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3 | ifnefalse 3901 |
. . . 4
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4 | 2, 3 | syl 16 |
. . 3
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5 | renemnf 9535 |
. . . 4
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6 | ifnefalse 3901 |
. . . 4
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7 | 5, 6 | syl 16 |
. . 3
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8 | 4, 7 | eqtrd 2492 |
. 2
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9 | 1, 8 | syl5eq 2504 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-resscn 9442 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-op 3984 df-uni 4192 df-br 4393 df-opab 4451 df-mpt 4452 df-id 4736 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-er 7203 df-en 7413 df-dom 7414 df-sdom 7415 df-pnf 9523 df-mnf 9524 df-xneg 11192 |
This theorem is referenced by: xneg0 11285 xnegcl 11286 xnegneg 11287 xltnegi 11289 rexsub 11306 xnegid 11309 xnegdi 11314 xpncan 11317 xnpcan 11318 xmulneg1 11335 xmulm1 11347 xadddi 11361 xlt2addrd 26187 xrsmulgzz 26275 |
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