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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 11300 | . . 3 ⊢ 𝐴 ∈ ℂ |
3 | resubcl.2 | . . . 4 ⊢ 𝐵 ∈ ℝ | |
4 | 3 | recni 11300 | . . 3 ⊢ 𝐵 ∈ ℂ |
5 | negsub 11580 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
6 | 2, 4, 5 | mp2an 691 | . 2 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
7 | 3 | renegcli 11593 | . . 3 ⊢ -𝐵 ∈ ℝ |
8 | 1, 7 | readdcli 11301 | . 2 ⊢ (𝐴 + -𝐵) ∈ ℝ |
9 | 6, 8 | eqeltrri 2835 | 1 ⊢ (𝐴 − 𝐵) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 (class class class)co 7445 ℂcc 11178 ℝcr 11179 + caddc 11183 − cmin 11516 -cneg 11517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-ltxr 11325 df-sub 11518 df-neg 11519 |
This theorem is referenced by: 0reALT 11629 emcllem7 27054 emre 27058 emgt0 27059 bposlem8 27344 chebbnd1lem3 27524 chebbnd1 27525 norm3adifii 31171 lnophmlem2 32040 dpmul4 32870 ballotlemi1 34459 logdivsqrle 34619 arearect 43116 areaquad 43117 stirlinglem13 45941 fouriersw 46086 |
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