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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 11389 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
2 | 1 | oveq2i 7369 | . . 3 ⊢ (𝐴 − -𝐵) = (𝐴 − (0 − 𝐵)) |
3 | 0cn 11148 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | subsub 11432 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (0 − 𝐵)) = ((𝐴 − 0) + 𝐵)) | |
5 | 3, 4 | mp3an2 1450 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (0 − 𝐵)) = ((𝐴 − 0) + 𝐵)) |
6 | 2, 5 | eqtrid 2789 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = ((𝐴 − 0) + 𝐵)) |
7 | subid1 11422 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) | |
8 | 7 | adantr 482 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 0) = 𝐴) |
9 | 8 | oveq1d 7373 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 0) + 𝐵) = (𝐴 + 𝐵)) |
10 | 6, 9 | eqtrd 2777 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 (class class class)co 7358 ℂcc 11050 0cc0 11052 + caddc 11055 − cmin 11386 -cneg 11387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-ltxr 11195 df-sub 11388 df-neg 11389 |
This theorem is referenced by: negneg 11452 negdi 11459 neg2sub 11462 subnegi 11481 subnegd 11520 recextlem1 11786 fzshftral 13530 shftval4 14963 sqreulem 15245 sqreu 15246 fsumshftm 15667 fsumcube 15944 eftlub 15992 summodnegmod 16170 shft2rab 24875 atandm2 26230 atandm4 26232 acosneg 26240 atanneg 26260 atancj 26263 atanlogadd 26267 atanlogsublem 26268 atanlogsub 26269 efiatan2 26270 2efiatan 26271 tanatan 26272 atans2 26284 dvatan 26288 atantayl 26290 wilthlem1 26420 wilthlem3 26422 wilthimp 26424 ftalem7 26431 ppiub 26555 2sqlem11 26780 2sqblem 26782 cos2h 36072 tan2h 36073 ftc1anclem5 36158 2pwp1prm 45788 |
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