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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > negsubdi3d | Structured version Visualization version GIF version |
Description: Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
negsubdi3d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
negsubdi3d.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
negsubdi3d | ⊢ (𝜑 → -(𝐴 − 𝐵) = (-𝐴 − -𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negsubdi3d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | negsubdi3d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | 1, 2 | negsubdi2d 10611 | . 2 ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
4 | 1, 2 | neg2subd 10612 | . 2 ⊢ (𝜑 → (-𝐴 − -𝐵) = (𝐵 − 𝐴)) |
5 | 3, 4 | eqtr4d 2808 | 1 ⊢ (𝜑 → -(𝐴 − 𝐵) = (-𝐴 − -𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 (class class class)co 6794 ℂcc 10137 − cmin 10469 -cneg 10470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-po 5171 df-so 5172 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-pnf 10279 df-mnf 10280 df-ltxr 10282 df-sub 10471 df-neg 10472 |
This theorem is referenced by: neglimc 40398 |
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