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Mirrors > Home > ILE Home > Th. List > resubcld | GIF version |
Description: Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
renegcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resubcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
resubcld | ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | resubcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | resubcl 7649 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 403 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 (class class class)co 5591 ℝcr 7252 − cmin 7556 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-setind 4316 ax-resscn 7340 ax-1cn 7341 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-sub 7558 df-neg 7559 |
This theorem is referenced by: ltsubadd 7813 lesubadd 7815 ltaddsub 7817 leaddsub 7819 lesub1 7837 lesub2 7838 ltsub1 7839 ltsub2 7840 lt2sub 7841 le2sub 7842 rereim 7963 ltmul1a 7968 cru 7979 lemul1a 8213 ztri3or 8689 lincmb01cmp 9315 iccf1o 9316 rebtwn2z 9555 qbtwnrelemcalc 9556 qbtwnre 9557 intfracq 9616 modqval 9620 modqlt 9629 modqsubdir 9689 serile 9790 expnbnd 9912 crre 10118 remullem 10132 recvguniqlem 10254 resqrexlemover 10270 resqrexlemcalc2 10275 resqrexlemcalc3 10276 resqrexlemnmsq 10277 resqrexlemnm 10278 resqrexlemcvg 10279 resqrexlemglsq 10282 resqrexlemga 10283 fzomaxdiflem 10372 caubnd2 10377 amgm2 10378 icodiamlt 10440 qdenre 10462 maxabslemab 10466 maxabslemlub 10467 maxltsup 10478 mulcn2 10525 climle 10546 climsqz 10547 climsqz2 10548 climcvg1nlem 10560 |
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