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Mathbox for Stefan O'Rear |
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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 21367 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
3 | 2 | subrgss 20576 | . . 3 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
5 | cnfldadd 21369 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
6 | 5 | subrgacl 20587 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) → (𝑎 + 𝑏) ∈ 𝑆) |
7 | 6 | 3expb 1118 | . . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
8 | 1, 7 | sylan 579 | . 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 20581 | . . 3 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld)) | |
12 | cnfld0 21404 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
13 | 12 | subg0cl 19150 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆) |
14 | 1, 11, 13 | 3syl 18 | . 2 ⊢ (𝜑 → 0 ∈ 𝑆) |
15 | 4, 8, 9, 10, 14 | fsumcllem 15754 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2104 ⊆ wss 3963 ‘cfv 6558 (class class class)co 7425 Fincfn 8978 ℂcc 11144 0cc0 11146 + caddc 11149 Σcsu 15708 SubGrpcsubg 19136 SubRingcsubrg 20571 ℂfldccnfld 21363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12518 df-z 12605 df-dec 12725 df-uz 12870 df-rp 13026 df-fz 13538 df-fzo 13682 df-seq 14029 df-exp 14089 df-hash 14356 df-cj 15124 df-re 15125 df-im 15126 df-sqrt 15260 df-abs 15261 df-clim 15510 df-sum 15709 df-struct 17170 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-starv 17302 df-tset 17306 df-ple 17307 df-ds 17309 df-unif 17310 df-0g 17477 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18952 df-subg 19139 df-cmn 19800 df-mgp 20138 df-ring 20238 df-cring 20239 df-subrg 20573 df-cnfld 21364 |
This theorem is referenced by: cnsrplycl 43110 |
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