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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 7943 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 406 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1461 (class class class)co 5726 ℝcr 7540 − cmin 7850 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-setind 4410 ax-resscn 7631 ax-1cn 7632 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-sub 7852 df-neg 7853 |
This theorem is referenced by: ltsubadd 8107 lesubadd 8109 ltaddsub 8111 leaddsub 8113 lesub1 8131 lesub2 8132 ltsub1 8133 ltsub2 8134 lt2sub 8135 le2sub 8136 rereim 8260 ltmul1a 8265 cru 8276 lemul1a 8520 ztri3or 8995 lincmb01cmp 9673 iccf1o 9674 rebtwn2z 9919 qbtwnrelemcalc 9920 qbtwnre 9921 intfracq 9980 modqval 9984 modqlt 9993 modqsubdir 10053 ser3le 10178 expnbnd 10302 crre 10516 remullem 10530 recvguniqlem 10652 resqrexlemover 10668 resqrexlemcalc2 10673 resqrexlemcalc3 10674 resqrexlemnmsq 10675 resqrexlemnm 10676 resqrexlemcvg 10677 resqrexlemglsq 10680 resqrexlemga 10681 fzomaxdiflem 10770 caubnd2 10775 amgm2 10776 icodiamlt 10838 qdenre 10860 maxabslemab 10864 maxabslemlub 10865 maxltsup 10876 bdtrilem 10896 bdtri 10897 mulcn2 10967 reccn2ap 10968 climle 10989 climsqz 10990 climsqz2 10991 climcvg1nlem 11004 fsumle 11118 cvgratnnlembern 11178 cvgratnnlemsumlt 11183 cvgratnnlemfm 11184 cvgratnnlemrate 11185 cvgratnn 11186 efltim 11249 sin01bnd 11309 sin01gt0 11313 blss2ps 12389 blss2 12390 blssps 12410 blss 12411 cvgcmp2nlemabs 12908 trilpolemlt1 12915 |
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