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Mirrors > Home > MPE Home > Th. List > subneg | Structured version Visualization version 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 10875 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
2 | 1 | oveq2i 7169 | . . 3 ⊢ (𝐴 − -𝐵) = (𝐴 − (0 − 𝐵)) |
3 | 0cn 10635 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | subsub 10918 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (0 − 𝐵)) = ((𝐴 − 0) + 𝐵)) | |
5 | 3, 4 | mp3an2 1445 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (0 − 𝐵)) = ((𝐴 − 0) + 𝐵)) |
6 | 2, 5 | syl5eq 2870 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = ((𝐴 − 0) + 𝐵)) |
7 | subid1 10908 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) | |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 0) = 𝐴) |
9 | 8 | oveq1d 7173 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 0) + 𝐵) = (𝐴 + 𝐵)) |
10 | 6, 9 | eqtrd 2858 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 0cc0 10539 + caddc 10542 − cmin 10872 -cneg 10873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 |
This theorem is referenced by: negneg 10938 negdi 10945 neg2sub 10948 subnegi 10967 subnegd 11006 recextlem1 11272 fzshftral 12998 shftval4 14438 sqreulem 14721 sqreu 14722 fsumshftm 15138 fsumcube 15416 eftlub 15464 summodnegmod 15642 shft2rab 24111 atandm2 25457 atandm4 25459 acosneg 25467 atanneg 25487 atancj 25490 atanlogadd 25494 atanlogsublem 25495 atanlogsub 25496 efiatan2 25497 2efiatan 25498 tanatan 25499 atans2 25511 dvatan 25515 atantayl 25517 wilthlem1 25647 wilthlem3 25649 wilthimp 25651 ftalem7 25658 ppiub 25782 2sqlem11 26007 2sqblem 26009 cos2h 34885 tan2h 34886 ftc1anclem5 34973 2pwp1prm 43758 |
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