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Mirrors > Home > MPE Home > Th. List > resubcli | Structured version Visualization version 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 10644 | . . 3 ⊢ 𝐴 ∈ ℂ |
3 | resubcl.2 | . . . 4 ⊢ 𝐵 ∈ ℝ | |
4 | 3 | recni 10644 | . . 3 ⊢ 𝐵 ∈ ℂ |
5 | negsub 10923 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
6 | 2, 4, 5 | mp2an 691 | . 2 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
7 | 3 | renegcli 10936 | . . 3 ⊢ -𝐵 ∈ ℝ |
8 | 1, 7 | readdcli 10645 | . 2 ⊢ (𝐴 + -𝐵) ∈ ℝ |
9 | 6, 8 | eqeltrri 2887 | 1 ⊢ (𝐴 − 𝐵) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 ℝcr 10525 + caddc 10529 − cmin 10859 -cneg 10860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 |
This theorem is referenced by: 0reALT 10972 emcllem7 25587 emre 25591 emgt0 25592 bposlem8 25875 chebbnd1lem3 26055 chebbnd1 26056 norm3adifii 28931 lnophmlem2 29800 dpmul4 30616 ballotlemi1 31870 logdivsqrle 32031 arearect 40165 areaquad 40166 stirlinglem13 42728 fouriersw 42873 |
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