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Mirrors > Home > ILE Home > Th. List > resubcli | GIF version |
Description: Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
renegcl.1 | ⊢ 𝐴 ∈ ℝ |
resubcl.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
resubcli | ⊢ (𝐴 − 𝐵) ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcl.1 | . . . 4 ⊢ 𝐴 ∈ ℝ | |
2 | 1 | recni 7988 | . . 3 ⊢ 𝐴 ∈ ℂ |
3 | resubcl.2 | . . . 4 ⊢ 𝐵 ∈ ℝ | |
4 | 3 | recni 7988 | . . 3 ⊢ 𝐵 ∈ ℂ |
5 | negsub 8224 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
6 | 2, 4, 5 | mp2an 426 | . 2 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
7 | 3 | renegcli 8238 | . . 3 ⊢ -𝐵 ∈ ℝ |
8 | 1, 7 | readdcli 7989 | . 2 ⊢ (𝐴 + -𝐵) ∈ ℝ |
9 | 6, 8 | eqeltrri 2263 | 1 ⊢ (𝐴 − 𝐵) ∈ ℝ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 (class class class)co 5891 ℂcc 7828 ℝcr 7829 + caddc 7833 − cmin 8147 -cneg 8148 |
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-in1 615 ax-in2 616 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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-setind 4551 ax-resscn 7922 ax-1cn 7923 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-distr 7934 ax-i2m1 7935 ax-0id 7938 ax-rnegex 7939 ax-cnre 7941 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-sub 8149 df-neg 8150 |
This theorem is referenced by: 0reALT 8273 |
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