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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 8178 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) | |
2 | negeu 8177 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) | |
3 | 2 | ancoms 268 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) |
4 | riotacl 5865 | . . 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 2266 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ∃!wreu 2470 ℩crio 5850 (class class class)co 5895 ℂcc 7838 + caddc 7843 − cmin 8157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-resscn 7932 ax-1cn 7933 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-distr 7944 ax-i2m1 7945 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-sub 8159 |
This theorem is referenced by: negcl 8186 subf 8188 pncan3 8194 npcan 8195 addsubass 8196 addsub 8197 addsub12 8199 addsubeq4 8201 npncan 8207 nppcan 8208 nnpcan 8209 nppcan3 8210 subcan2 8211 subsub2 8214 subsub4 8219 nnncan 8221 nnncan1 8222 nnncan2 8223 npncan3 8224 addsub4 8229 subadd4 8230 peano2cnm 8252 subcli 8262 subcld 8297 subeqrev 8362 subdi 8371 subdir 8372 mulsub2 8388 recextlem1 8637 recexap 8639 div2subap 8823 cju 8947 halfaddsubcl 9181 halfaddsub 9182 iccf1o 10033 ser3sub 10536 sqsubswap 10610 subsq 10657 subsq2 10658 bcn2 10775 shftval2 10866 2shfti 10871 sqabssub 11096 abssub 11141 abs3dif 11145 abs2dif 11146 abs2difabs 11148 climuni 11332 cjcn2 11355 recn2 11356 imcn2 11357 climsub 11367 fisum0diag2 11486 arisum2 11538 geosergap 11545 geolim 11550 geolim2 11551 georeclim 11552 geo2sum 11553 tanaddap 11778 addsin 11781 fzocongeq 11895 odd2np1 11909 phiprm 12254 pythagtriplem4 12299 pythagtriplem12 12306 pythagtriplem14 12308 fldivp1 12379 4sqlem19 12440 cnmet 14482 dveflem 14639 dvef 14640 efimpi 14692 ptolemy 14697 tangtx 14711 abssinper 14719 |
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