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Mirrors > Home > MPE Home > Th. List > fsumcllem | Structured version Visualization version GIF version |
Description: - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.) |
Ref | Expression |
---|---|
fsumcllem.1 | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
fsumcllem.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
fsumcllem.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumcllem.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
fsumcllem.5 | ⊢ (𝜑 → 0 ∈ 𝑆) |
Ref | Expression |
---|---|
fsumcllem | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 𝐴 = ∅) | |
2 | 1 | sumeq1d 15683 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵) |
3 | sum0 15703 | . . . 4 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0 | |
4 | 2, 3 | eqtrdi 2781 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
5 | fsumcllem.5 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝑆) | |
6 | 5 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 0 ∈ 𝑆) |
7 | 4, 6 | eqeltrd 2825 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
8 | fsumcllem.1 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
9 | 8 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝑆 ⊆ ℂ) |
10 | fsumcllem.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
11 | 10 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
12 | fsumcllem.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
13 | 12 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) |
14 | fsumcllem.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
15 | 14 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
16 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
17 | 9, 11, 13, 15, 16 | fsumcl2lem 15713 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
18 | 7, 17 | pm2.61dane 3018 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ⊆ wss 3944 ∅c0 4322 (class class class)co 7419 Fincfn 8964 ℂcc 11138 0cc0 11140 + caddc 11143 Σcsu 15668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 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 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-sum 15669 |
This theorem is referenced by: fsumcl 15715 fsumrecl 15716 fsumzcl 15717 fsumnn0cl 15718 fsumge0 15777 plymullem 26195 efnnfsumcl 27080 efchtdvds 27136 fsumrp0cl 32840 fsumcnsrcl 42732 aacllem 48420 |
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