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Mirrors > Home > ILE Home > Th. List > subcl | GIF version |
Description: Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.) |
Ref | Expression |
---|---|
subcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subval 8098 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) | |
2 | negeu 8097 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) | |
3 | 2 | ancoms 266 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) |
4 | riotacl 5820 | . . 3 ⊢ (∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴 → (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ) | |
5 | 3, 4 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ) |
6 | 1, 5 | eqeltrd 2247 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 ∃!wreu 2450 ℩crio 5805 (class class class)co 5850 ℂcc 7759 + caddc 7764 − cmin 8077 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7853 ax-1cn 7854 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-sub 8079 |
This theorem is referenced by: negcl 8106 subf 8108 pncan3 8114 npcan 8115 addsubass 8116 addsub 8117 addsub12 8119 addsubeq4 8121 npncan 8127 nppcan 8128 nnpcan 8129 nppcan3 8130 subcan2 8131 subsub2 8134 subsub4 8139 nnncan 8141 nnncan1 8142 nnncan2 8143 npncan3 8144 addsub4 8149 subadd4 8150 peano2cnm 8172 subcli 8182 subcld 8217 subeqrev 8282 subdi 8291 subdir 8292 mulsub2 8308 recextlem1 8556 recexap 8558 div2subap 8741 cju 8864 halfaddsubcl 9098 halfaddsub 9099 iccf1o 9948 ser3sub 10449 sqsubswap 10523 subsq 10569 subsq2 10570 bcn2 10685 shftval2 10777 2shfti 10782 sqabssub 11007 abssub 11052 abs3dif 11056 abs2dif 11057 abs2difabs 11059 climuni 11243 cjcn2 11266 recn2 11267 imcn2 11268 climsub 11278 fisum0diag2 11397 arisum2 11449 geosergap 11456 geolim 11461 geolim2 11462 georeclim 11463 geo2sum 11464 tanaddap 11689 addsin 11692 fzocongeq 11805 odd2np1 11819 phiprm 12164 pythagtriplem4 12209 pythagtriplem12 12216 pythagtriplem14 12218 fldivp1 12287 cnmet 13245 dveflem 13402 dvef 13403 efimpi 13455 ptolemy 13460 tangtx 13474 abssinper 13482 |
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