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Mirrors > Home > ILE Home > Th. List > mvrraddi | GIF version |
Description: Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
mvrraddi.1 | ⊢ 𝐵 ∈ ℂ |
mvrraddi.2 | ⊢ 𝐶 ∈ ℂ |
mvrraddi.3 | ⊢ 𝐴 = (𝐵 + 𝐶) |
Ref | Expression |
---|---|
mvrraddi | ⊢ (𝐴 − 𝐶) = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrraddi.3 | . . 3 ⊢ 𝐴 = (𝐵 + 𝐶) | |
2 | 1 | oveq1i 5784 | . 2 ⊢ (𝐴 − 𝐶) = ((𝐵 + 𝐶) − 𝐶) |
3 | mvrraddi.1 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | mvrraddi.2 | . . 3 ⊢ 𝐶 ∈ ℂ | |
5 | 3, 4 | pncan3oi 7981 | . 2 ⊢ ((𝐵 + 𝐶) − 𝐶) = 𝐵 |
6 | 2, 5 | eqtri 2160 | 1 ⊢ (𝐴 − 𝐶) = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7621 + caddc 7626 − cmin 7936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7715 ax-1cn 7716 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7938 |
This theorem is referenced by: 4m1e3 8844 5m1e4 8845 6m1e5 8846 7m1e6 8847 8m1e7 8848 9m1e8 8849 10m1e9 9280 fldiv4p1lem1div2 10081 |
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