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Mirrors > Home > ILE Home > Th. List > pnncani | GIF version |
Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
pncan3i.2 | ⊢ 𝐵 ∈ ℂ |
subadd.3 | ⊢ 𝐶 ∈ ℂ |
Ref | Expression |
---|---|
pnncani | ⊢ ((𝐴 + 𝐵) − (𝐴 − 𝐶)) = (𝐵 + 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | pncan3i.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | subadd.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | pnncan 8117 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 − 𝐶)) = (𝐵 + 𝐶)) | |
5 | 1, 2, 3, 4 | mp3an 1319 | 1 ⊢ ((𝐴 + 𝐵) − (𝐴 − 𝐶)) = (𝐵 + 𝐶) |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 (class class class)co 5825 ℂcc 7731 + caddc 7736 − cmin 8047 |
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 4083 ax-pow 4136 ax-pr 4170 ax-setind 4497 ax-resscn 7825 ax-1cn 7826 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 |
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 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-sub 8049 |
This theorem is referenced by: (None) |
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