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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 7771 | . . 3 ⊢ 𝐴 ∈ ℂ |
3 | resubcl.2 | . . . 4 ⊢ 𝐵 ∈ ℝ | |
4 | 3 | recni 7771 | . . 3 ⊢ 𝐵 ∈ ℂ |
5 | negsub 8003 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
6 | 2, 4, 5 | mp2an 422 | . 2 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
7 | 3 | renegcli 8017 | . . 3 ⊢ -𝐵 ∈ ℝ |
8 | 1, 7 | readdcli 7772 | . 2 ⊢ (𝐴 + -𝐵) ∈ ℝ |
9 | 6, 8 | eqeltrri 2211 | 1 ⊢ (𝐴 − 𝐵) ∈ ℝ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5767 ℂcc 7611 ℝcr 7612 + caddc 7616 − cmin 7926 -cneg 7927 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-neg 7929 |
This theorem is referenced by: 0reALT 8052 |
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