![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumcnsrcl | Structured version Visualization version GIF version |
Description: Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
Ref | Expression |
---|---|
fsumcnsrcl.s | ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) |
fsumcnsrcl.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumcnsrcl.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
Ref | Expression |
---|---|
fsumcnsrcl | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumcnsrcl.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) | |
2 | cnfldbas 21360 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
3 | 2 | subrgss 20564 | . . 3 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
5 | cnfldadd 21362 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
6 | 5 | subrgacl 20575 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) → (𝑎 + 𝑏) ∈ 𝑆) |
7 | 6 | 3expb 1121 | . . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
8 | 1, 7 | sylan 580 | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
9 | fsumcnsrcl.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
10 | fsumcnsrcl.b | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
11 | subrgsubg 20569 | . . 3 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld)) | |
12 | cnfld0 21397 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
13 | 12 | subg0cl 19148 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆) |
14 | 1, 11, 13 | 3syl 18 | . 2 ⊢ (𝜑 → 0 ∈ 𝑆) |
15 | 4, 8, 9, 10, 14 | fsumcllem 15764 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3950 ‘cfv 6559 (class class class)co 7429 Fincfn 8981 ℂcc 11149 0cc0 11151 + caddc 11154 Σcsu 15718 SubGrpcsubg 19134 SubRingcsubrg 20561 ℂfldccnfld 21356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-inf2 9677 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 ax-addf 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-rp 13031 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-starv 17308 df-tset 17312 df-ple 17313 df-ds 17315 df-unif 17316 df-0g 17482 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-grp 18950 df-subg 19137 df-cmn 19796 df-mgp 20134 df-ring 20228 df-cring 20229 df-subrg 20562 df-cnfld 21357 |
This theorem is referenced by: cnsrplycl 43157 |
Copyright terms: Public domain | W3C validator |