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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 8155 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) | |
5 | 1, 2, 3, 4 | mp3an 1348 | 1 ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∈ wcel 2160 (class class class)co 5891 ℂcc 7827 + caddc 7832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-resscn 7921 ax-1cn 7922 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-distr 7933 ax-i2m1 7934 ax-0id 7937 ax-rnegex 7938 ax-cnre 7940 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5193 df-fv 5239 df-ov 5894 |
This theorem is referenced by: negdii 8259 fsumrelem 11497 |
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