![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mvrladdd | Structured version Visualization version GIF version |
Description: Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
mvrraddd.1 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mvrraddd.2 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
mvrraddd.3 | ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) |
Ref | Expression |
---|---|
mvrladdd | ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrraddd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
2 | mvrraddd.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mvrraddd.3 | . . 3 ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) | |
4 | 2, 1, 3 | comraddd 10590 | . 2 ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) |
5 | 1, 2, 4 | mvrraddd 10787 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = 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: 2txmxeqx 11522 cvgcmpce 14954 mertens 15021 sin01bnd 15317 cos01bnd 15318 eirrlem 15336 bitsmod 15564 tangtx 24695 quart1lem 25033 efiatan2 25095 log2tlbnd 25124 jensenlem2 25166 fsumharmonic 25190 chtublem 25388 bcctr 25452 pcbcctr 25453 bcp1ctr 25456 bposlem9 25469 lgsquadlem1 25557 selberg2lem 25691 logdivbnd 25697 pntrsumo1 25706 pntrsumbnd2 25708 pntrlog2bndlem6 25724 pntpbnd1a 25726 hgt750lemd 31328 bcprod 32218 dnizphlfeqhlf 33049 jm3.1lem1 38543 fzisoeu 40423 supxrgelem 40461 sigarcol 41980 dignn0flhalflem1 43424 1subrec1sub 43441 i2linesd 43631 |
Copyright terms: Public domain | W3C validator |