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Mirrors > Home > MPE Home > Th. List > negsubi | Structured version Visualization version Unicode version |
Description: Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
negidi.1 |
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pncan3i.2 |
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Ref | Expression |
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negsubi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 |
. 2
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2 | pncan3i.2 |
. 2
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3 | negsub 9922 |
. 2
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4 | 1, 2, 3 | mp2an 678 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-po 4755 df-so 4756 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 df-pnf 9677 df-mnf 9678 df-ltxr 9680 df-sub 9862 df-neg 9863 |
This theorem is referenced by: negsubdii 9960 negsubdi2i 9961 crreczi 12397 modsubi 15044 cosq14gt0 23465 cosq14ge0 23466 cosne0 23479 resinf1o 23485 eff1o 23498 1cubrlem 23767 acosbnd 23826 atanlogsublem 23841 dvatan 23861 log2cnv 23870 lgsdir2lem1 24251 ax5seglem7 24965 normlem9 26771 normpari 26807 polid2i 26810 lnophmlem2 27670 ballotlem2 29321 quad3 30302 cosnegpi 37742 dirkertrigeqlem3 37962 3exp4mod41 38916 zlmodzxzequap 40345 |
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