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Mirrors > Home > ILE Home > Th. List > addcani | GIF version |
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
addcani.1 | ⊢ 𝐴 ∈ ℂ |
addcani.2 | ⊢ 𝐵 ∈ ℂ |
addcani.3 | ⊢ 𝐶 ∈ ℂ |
Ref | Expression |
---|---|
addcani | ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcani.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | addcani.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | addcani.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | addcan 7910 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) | |
5 | 1, 2, 3, 4 | mp3an 1300 | 1 ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 + caddc 7591 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: negdii 8014 fsumrelem 11208 |
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