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Mirrors > Home > ILE Home > Th. List > pnpcan | GIF version |
Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
pnpcan | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 − 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 939 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ) | |
2 | simp2 940 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ) | |
3 | 1, 2 | addcld 7200 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
4 | simp3 941 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ) | |
5 | subsub4 7408 | . . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 + 𝐵) − 𝐴) − 𝐶) = ((𝐴 + 𝐵) − (𝐴 + 𝐶))) | |
6 | 3, 1, 4, 5 | syl3anc 1170 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 + 𝐵) − 𝐴) − 𝐶) = ((𝐴 + 𝐵) − (𝐴 + 𝐶))) |
7 | pncan2 7382 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵) | |
8 | 7 | 3adant3 959 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵) |
9 | 8 | oveq1d 5558 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 + 𝐵) − 𝐴) − 𝐶) = (𝐵 − 𝐶)) |
10 | 6, 9 | eqtr3d 2116 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 − 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 (class class class)co 5543 ℂcc 7041 + caddc 7046 − cmin 7346 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-setind 4288 ax-resscn 7130 ax-1cn 7131 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-addcom 7138 ax-addass 7140 ax-distr 7142 ax-i2m1 7143 ax-0id 7146 ax-rnegex 7147 ax-cnre 7149 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-id 4056 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-iota 4897 df-fun 4934 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-sub 7348 |
This theorem is referenced by: pnpcan2 7415 pnpcand 7523 cju 8105 |
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