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Mirrors > Home > ILE Home > Th. List > subneg | GIF version |
Description: Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
subneg | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 8028 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
2 | 1 | oveq2i 5825 | . . 3 ⊢ (𝐴 − -𝐵) = (𝐴 − (0 − 𝐵)) |
3 | 0cn 7849 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | subsub 8084 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (0 − 𝐵)) = ((𝐴 − 0) + 𝐵)) | |
5 | 3, 4 | mp3an2 1304 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (0 − 𝐵)) = ((𝐴 − 0) + 𝐵)) |
6 | 2, 5 | syl5eq 2199 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = ((𝐴 − 0) + 𝐵)) |
7 | subid1 8074 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) | |
8 | 7 | adantr 274 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 0) = 𝐴) |
9 | 8 | oveq1d 5829 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 0) + 𝐵) = (𝐴 + 𝐵)) |
10 | 6, 9 | eqtrd 2187 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 2125 (class class class)co 5814 ℂcc 7709 0cc0 7711 + caddc 7714 − cmin 8025 -cneg 8026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-setind 4490 ax-resscn 7803 ax-1cn 7804 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-addcom 7811 ax-addass 7813 ax-distr 7815 ax-i2m1 7816 ax-0id 7819 ax-rnegex 7820 ax-cnre 7822 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-ral 2437 df-rex 2438 df-reu 2439 df-rab 2441 df-v 2711 df-sbc 2934 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-br 3962 df-opab 4022 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-iota 5128 df-fun 5165 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-sub 8027 df-neg 8028 |
This theorem is referenced by: negneg 8104 negdi 8111 neg2sub 8114 subnegi 8133 subnegd 8172 recextlem1 8504 fzshftral 9988 shftval4 10705 fsumshftm 11319 eftlub 11564 summodnegmod 11691 |
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