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Mirrors > Home > ILE Home > Th. List > addcan2i | GIF version |
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
addcani.1 | ⊢ 𝐴 ∈ ℂ |
addcani.2 | ⊢ 𝐵 ∈ ℂ |
addcani.3 | ⊢ 𝐶 ∈ ℂ |
Ref | Expression |
---|---|
addcan2i | ⊢ ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcani.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | addcani.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | addcani.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | addcan2 7192 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) | |
5 | 1, 2, 3, 4 | mp3an 1232 | 1 ⊢ ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 ℂcc 6887 + caddc 6892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: (None) |
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