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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 5837 | . 2 ⊢ (𝐴 − 𝐶) = ((𝐵 + 𝐶) − 𝐶) |
3 | mvrraddi.1 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | mvrraddi.2 | . . 3 ⊢ 𝐶 ∈ ℂ | |
5 | 3, 4 | pncan3oi 8096 | . 2 ⊢ ((𝐵 + 𝐶) − 𝐶) = 𝐵 |
6 | 2, 5 | eqtri 2178 | 1 ⊢ (𝐴 − 𝐶) = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 (class class class)co 5827 ℂcc 7733 + caddc 7738 − cmin 8051 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-setind 4499 ax-resscn 7827 ax-1cn 7828 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-sub 8053 |
This theorem is referenced by: 4m1e3 8960 5m1e4 8961 6m1e5 8962 7m1e6 8963 8m1e7 8964 9m1e8 8965 10m1e9 9396 fldiv4p1lem1div2 10214 |
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