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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 8162 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2136 (class class class)co 5842 ℝcr 7752 − cmin 8069 |
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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-setind 4514 ax-resscn 7845 ax-1cn 7846 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-addass 7855 ax-distr 7857 ax-i2m1 7858 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-sub 8071 df-neg 8072 |
This theorem is referenced by: ltsubadd 8330 lesubadd 8332 ltaddsub 8334 leaddsub 8336 lesub1 8354 lesub2 8355 ltsub1 8356 ltsub2 8357 lt2sub 8358 le2sub 8359 rereim 8484 ltmul1a 8489 cru 8500 lemul1a 8753 ztri3or 9234 lincmb01cmp 9939 iccf1o 9940 rebtwn2z 10190 qbtwnrelemcalc 10191 qbtwnre 10192 intfracq 10255 modqval 10259 modqlt 10268 modqsubdir 10328 ser3le 10453 expnbnd 10578 crre 10799 remullem 10813 recvguniqlem 10936 resqrexlemover 10952 resqrexlemcalc2 10957 resqrexlemcalc3 10958 resqrexlemnmsq 10959 resqrexlemnm 10960 resqrexlemcvg 10961 resqrexlemglsq 10964 resqrexlemga 10965 fzomaxdiflem 11054 caubnd2 11059 amgm2 11060 icodiamlt 11122 qdenre 11144 maxabslemab 11148 maxabslemlub 11149 maxltsup 11160 bdtrilem 11180 bdtri 11181 mulcn2 11253 reccn2ap 11254 climle 11275 climsqz 11276 climsqz2 11277 climcvg1nlem 11290 fsumle 11404 cvgratnnlembern 11464 cvgratnnlemsumlt 11469 cvgratnnlemfm 11470 cvgratnnlemrate 11471 cvgratnn 11472 efltim 11639 sin01bnd 11698 sin01gt0 11702 cos12dec 11708 uzwodc 11970 pythagtriplem12 12207 pythagtriplem14 12209 blss2ps 13046 blss2 13047 blssps 13067 blss 13068 ivthinclemlopn 13254 ivthinclemuopn 13256 dvcjbr 13312 reeff1oleme 13333 efltlemlt 13335 sin0pilem1 13342 tangtx 13399 cosordlem 13410 cosq34lt1 13411 cvgcmp2nlemabs 13911 iooref1o 13913 trilpolemlt1 13920 trirec0 13923 apdifflemf 13925 |
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