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| Mirrors > Home > MPE Home > Th. List > subadd2d | Structured version Visualization version GIF version | ||
| Description: Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| subaddd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| Ref | Expression |
|---|---|
| subadd2d | ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | subaddd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 4 | subadd2 11512 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 + caddc 11158 − cmin 11492 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 |
| This theorem is referenced by: addeq0 11686 icoshftf1o 13514 iccf1o 13536 modmuladdnn0 13956 hashun3 14423 oddm1even 16380 oexpneg 16382 modremain 16445 hashdvds 16812 psgnunilem5 19512 icopnfcnv 24973 affineequiv4 26869 mcubic 26890 lgsvalmod 27360 2sqmod 27480 colinearalglem2 28922 wlklnwwlkln2lem 29902 eucrct2eupth 30264 ballotlem1c 34510 subfacp1lem1 35184 mblfinlem3 37666 mblfinlem4 37667 itg2addnclem2 37679 aks4d1p1p7 42075 aks4d1p1 42077 fperdvper 45934 fourierdlem19 46141 fmtnorec2lem 47529 fmtnorec4 47536 fmtnoprmfac1lem 47551 fmtnoprmfac1 47552 fmtnoprmfac2 47554 sfprmdvdsmersenne 47590 oexpnegALTV 47664 even3prm2 47706 sbgoldbst 47765 nnsgrpnmnd 48094 blennn0em1 48512 eenglngeehlnmlem1 48658 eenglngeehlnmlem2 48659 itscnhlc0xyqsol 48686 itschlc0xyqsol1 48687 |
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