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Mirrors > Home > ILE Home > Th. List > negneg | GIF version |
Description: A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negneg | ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 7349 | . . 3 ⊢ --𝐴 = (0 − -𝐴) | |
2 | 0cn 7173 | . . . 4 ⊢ 0 ∈ ℂ | |
3 | subneg 7424 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 − -𝐴) = (0 + 𝐴)) | |
4 | 2, 3 | mpan 415 | . . 3 ⊢ (𝐴 ∈ ℂ → (0 − -𝐴) = (0 + 𝐴)) |
5 | 1, 4 | syl5eq 2126 | . 2 ⊢ (𝐴 ∈ ℂ → --𝐴 = (0 + 𝐴)) |
6 | addid2 7314 | . 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) | |
7 | 5, 6 | eqtrd 2114 | 1 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 (class class class)co 5543 ℂcc 7041 0cc0 7043 + caddc 7046 − cmin 7346 -cneg 7347 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-setind 4288 ax-resscn 7130 ax-1cn 7131 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-addcom 7138 ax-addass 7140 ax-distr 7142 ax-i2m1 7143 ax-0id 7146 ax-rnegex 7147 ax-cnre 7149 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-id 4056 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-iota 4897 df-fun 4934 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-sub 7348 df-neg 7349 |
This theorem is referenced by: neg11 7426 negcon1 7427 negreb 7440 negnegi 7445 negnegd 7477 negf1o 7553 mul2neg 7569 divneg2ap 7891 nnnegz 8435 znegclb 8465 expineg2 9582 shftcan2 9861 negfi 10248 dvdsnegb 10357 |
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