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Mirrors > Home > ILE Home > Th. List > resubcl | GIF version |
Description: Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.) |
Ref | Expression |
---|---|
resubcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7777 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | recn 7777 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
3 | negsub 8034 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
4 | 1, 2, 3 | syl2an 287 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
5 | renegcl 8047 | . . 3 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
6 | readdcl 7770 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → (𝐴 + -𝐵) ∈ ℝ) | |
7 | 5, 6 | sylan2 284 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + -𝐵) ∈ ℝ) |
8 | 4, 7 | eqeltrrd 2218 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 ℝcr 7643 + caddc 7647 − cmin 7957 -cneg 7958 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-neg 7960 |
This theorem is referenced by: peano2rem 8053 resubcld 8167 posdif 8241 lt2sub 8246 le2sub 8247 cju 8743 elz2 9146 difrp 9509 iooshf 9765 iccshftl 9809 lincmb01cmp 9816 uzsubsubfz 9858 difelfzle 9942 fzonmapblen 9995 eluzgtdifelfzo 10005 subfzo0 10050 modfzo0difsn 10199 expubnd 10381 absdiflt 10896 absdifle 10897 elicc4abs 10898 abssubge0 10906 abs2difabs 10912 maxabsle 11008 resin4p 11461 recos4p 11462 cos01bnd 11501 cos01gt0 11505 bl2ioo 12750 ioo2bl 12751 ioo2blex 12752 blssioo 12753 sincosq1sgn 12955 sincosq2sgn 12956 sincosq3sgn 12957 sincosq4sgn 12958 sinq12gt0 12959 cosq14gt0 12961 tangtx 12967 relogdiv 12999 logdivlti 13010 |
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