Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mulm1i | Structured version Visualization version GIF version |
Description: Product with minus one is negative. (Contributed by NM, 31-Jul-1999.) |
Ref | Expression |
---|---|
mulm1.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
mulm1i | ⊢ (-1 · 𝐴) = -𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mulm1 11399 | . 2 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (-1 · 𝐴) = -𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2109 (class class class)co 7268 ℂcc 10853 1c1 10856 · cmul 10860 -cneg 11189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-ltxr 10998 df-sub 11190 df-neg 11191 |
This theorem is referenced by: i3 13901 evpmodpmf1o 20782 efif1olem2 25680 logf1o2 25786 tanatan 26050 lgsneg 26450 lgsdilem 26453 lgsdir2lem5 26458 ipval3 29050 ipasslem10 29180 normlem0 29450 normlem9 29459 polid2i 29498 quad3 33607 sqrtcval2 41203 cosnegpi 43362 sqwvfoura 43723 |
Copyright terms: Public domain | W3C validator |