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Mirrors > Home > ILE Home > Th. List > subcld | GIF version |
Description: Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
subcld | ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | subcl 7929 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 − cmin 7901 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 |
This theorem is referenced by: pnpncand 8105 kcnktkm1cn 8113 muleqadd 8397 peano2zm 9060 peano5uzti 9127 modqmuladdnn0 10109 modsumfzodifsn 10137 hashfz 10535 hashfzo 10536 shftfvalg 10558 ovshftex 10559 shftfibg 10560 shftfval 10561 shftdm 10562 shftfib 10563 shftval 10565 2shfti 10571 crre 10597 remim 10600 remullem 10611 resqrexlemover 10750 resqrexlemcalc1 10754 abssubne0 10831 abs3lem 10851 caubnd2 10857 maxabslemlub 10947 maxabslemval 10948 maxcl 10950 minabs 10975 bdtrilem 10978 bdtri 10979 climuni 11030 mulcn2 11049 reccn2ap 11050 cn1lem 11051 climcvg1nlem 11086 fsumparts 11207 arisum2 11236 geosergap 11243 geo2sum2 11252 geoisum1c 11257 cvgratnnlemrate 11267 sinval 11336 sinf 11338 tanval2ap 11347 tanval3ap 11348 sinneg 11360 efival 11366 cos12dec 11401 addcncntoplem 12647 mulcncflem 12686 cnopnap 12690 limcimolemlt 12729 limcimo 12730 cnplimclemle 12733 limccnp2lem 12741 dvlemap 12745 dvconst 12757 dvid 12758 dvcnp2cntop 12759 dvaddxxbr 12761 dvmulxxbr 12762 dvcoapbr 12767 dvcjbr 12768 dvrecap 12773 dveflem 12782 dvef 12783 sin0pilem1 12789 ptolemy 12832 tangtx 12846 cosq34lt1 12858 qdencn 13149 |
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