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Mirrors > Home > ILE Home > Th. List > subsub2 | GIF version |
Description: Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
subsub2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = (𝐴 + (𝐶 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcl 7961 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 − 𝐶) ∈ ℂ) | |
2 | 1 | 3adant1 999 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 − 𝐶) ∈ ℂ) |
3 | simp1 981 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ) | |
4 | simp3 983 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ) | |
5 | simp2 982 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ) | |
6 | subcl 7961 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ) | |
7 | 4, 5, 6 | syl2anc 408 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ) |
8 | 2, 3, 7 | add12d 7929 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 − 𝐶) + (𝐴 + (𝐶 − 𝐵))) = (𝐴 + ((𝐵 − 𝐶) + (𝐶 − 𝐵)))) |
9 | npncan2 7989 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 − 𝐶) + (𝐶 − 𝐵)) = 0) | |
10 | 9 | 3adant1 999 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 − 𝐶) + (𝐶 − 𝐵)) = 0) |
11 | 10 | oveq2d 5790 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + ((𝐵 − 𝐶) + (𝐶 − 𝐵))) = (𝐴 + 0)) |
12 | 3 | addid1d 7911 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + 0) = 𝐴) |
13 | 8, 11, 12 | 3eqtrd 2176 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐵 − 𝐶) + (𝐴 + (𝐶 − 𝐵))) = 𝐴) |
14 | 3, 7 | addcld 7785 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐶 − 𝐵)) ∈ ℂ) |
15 | subadd 7965 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 − 𝐶) ∈ ℂ ∧ (𝐴 + (𝐶 − 𝐵)) ∈ ℂ) → ((𝐴 − (𝐵 − 𝐶)) = (𝐴 + (𝐶 − 𝐵)) ↔ ((𝐵 − 𝐶) + (𝐴 + (𝐶 − 𝐵))) = 𝐴)) | |
16 | 3, 2, 14, 15 | syl3anc 1216 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 − 𝐶)) = (𝐴 + (𝐶 − 𝐵)) ↔ ((𝐵 − 𝐶) + (𝐴 + (𝐶 − 𝐵))) = 𝐴)) |
17 | 13, 16 | mpbird 166 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = (𝐴 + (𝐶 − 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 0cc0 7620 + caddc 7623 − cmin 7933 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 |
This theorem is referenced by: nncan 7991 subsub 7992 subsub3 7994 ppncan 8004 subadd4 8006 subsub2d 8102 |
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