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Mirrors > Home > ILE Home > Th. List > addsubass | GIF version |
Description: Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addsubass | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 939 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ) | |
2 | subcl 7426 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 − 𝐶) ∈ ℂ) | |
3 | 2 | 3adant1 957 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 − 𝐶) ∈ ℂ) |
4 | simp3 941 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ) | |
5 | 1, 3, 4 | addassd 7255 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + (𝐵 − 𝐶)) + 𝐶) = (𝐴 + ((𝐵 − 𝐶) + 𝐶))) |
6 | npcan 7436 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 − 𝐶) + 𝐶) = 𝐵) | |
7 | 6 | 3adant1 957 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 − 𝐶) + 𝐶) = 𝐵) |
8 | 7 | oveq2d 5579 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + ((𝐵 − 𝐶) + 𝐶)) = (𝐴 + 𝐵)) |
9 | 5, 8 | eqtrd 2115 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + (𝐵 − 𝐶)) + 𝐶) = (𝐴 + 𝐵)) |
10 | 9 | oveq1d 5578 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 + (𝐵 − 𝐶)) + 𝐶) − 𝐶) = ((𝐴 + 𝐵) − 𝐶)) |
11 | 1, 3 | addcld 7252 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 − 𝐶)) ∈ ℂ) |
12 | pncan 7433 | . . 3 ⊢ (((𝐴 + (𝐵 − 𝐶)) ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 + (𝐵 − 𝐶)) + 𝐶) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) | |
13 | 11, 4, 12 | syl2anc 403 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 + (𝐵 − 𝐶)) + 𝐶) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) |
14 | 10, 13 | eqtr3d 2117 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 (class class class)co 5563 ℂcc 7093 + caddc 7098 − cmin 7398 |
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 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-setind 4308 ax-resscn 7182 ax-1cn 7183 ax-icn 7185 ax-addcl 7186 ax-addrcl 7187 ax-mulcl 7188 ax-addcom 7190 ax-addass 7192 ax-distr 7194 ax-i2m1 7195 ax-0id 7198 ax-rnegex 7199 ax-cnre 7201 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-br 3806 df-opab 3860 df-id 4076 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-iota 4917 df-fun 4954 df-fv 4960 df-riota 5519 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-sub 7400 |
This theorem is referenced by: addsub 7438 subadd23 7439 addsubeq4 7442 npncan 7448 subsub 7457 subsub3 7459 addsub4 7470 negsub 7475 addsubassi 7518 addsubassd 7558 zeo 8585 frecfzen2 9561 odd2np1 10480 |
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