![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subaddrii | Structured version Visualization version GIF version |
Description: Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
pncan3i.2 | ⊢ 𝐵 ∈ ℂ |
subadd.3 | ⊢ 𝐶 ∈ ℂ |
subaddri.4 | ⊢ (𝐵 + 𝐶) = 𝐴 |
Ref | Expression |
---|---|
subaddrii | ⊢ (𝐴 − 𝐵) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subaddri.4 | . 2 ⊢ (𝐵 + 𝐶) = 𝐴 | |
2 | negidi.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | pncan3i.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | subadd.3 | . . 3 ⊢ 𝐶 ∈ ℂ | |
5 | 2, 3, 4 | subaddi 10710 | . 2 ⊢ ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴) |
6 | 1, 5 | mpbir 223 | 1 ⊢ (𝐴 − 𝐵) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 + caddc 10275 − cmin 10606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 |
This theorem is referenced by: 2m1e1 11508 halfthird 11990 5recm6rec 11991 4bc2eq6 13434 bpoly3 15191 bpoly4 15192 cos1bnd 15319 cos2bnd 15320 pythagtriplem1 15925 iblitg 23972 cosq14gt0 24700 cosq14ge0 24701 sincos6thpi 24705 pige3 24707 cosne0 24714 resinf1o 24720 logimul 24797 ang180lem2 24988 mcubic 25025 quartlem1 25035 acosneg 25065 acosbnd 25078 atanlogsublem 25093 chtub 25389 lgsdir2lem1 25502 lgsdir2lem2 25503 lgsdir2lem3 25504 addltmulALT 29877 fib5 31066 fib6 31067 hgt750lem 31331 problem3 32158 problem4 32159 lhe4.4ex1a 39488 stoweidlem13 41161 stoweidlem26 41174 wallispilem4 41216 41prothprmlem2 42560 linevalexample 43203 5m4e1 43653 |
Copyright terms: Public domain | W3C validator |