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Mirrors > Home > MPE Home > Th. List > fsumre | Structured version Visualization version GIF version |
Description: The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.) |
Ref | Expression |
---|---|
fsumre.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumre.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
fsumre | ⊢ (𝜑 → (ℜ‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (ℜ‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumre.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fsumre.2 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
3 | ref 15054 | . . 3 ⊢ ℜ:ℂ⟶ℝ | |
4 | ax-resscn 11162 | . . 3 ⊢ ℝ ⊆ ℂ | |
5 | fss 6730 | . . 3 ⊢ ((ℜ:ℂ⟶ℝ ∧ ℝ ⊆ ℂ) → ℜ:ℂ⟶ℂ) | |
6 | 3, 4, 5 | mp2an 691 | . 2 ⊢ ℜ:ℂ⟶ℂ |
7 | readd 15068 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (ℜ‘(𝑥 + 𝑦)) = ((ℜ‘𝑥) + (ℜ‘𝑦))) | |
8 | 1, 2, 6, 7 | fsumrelem 15748 | 1 ⊢ (𝜑 → (ℜ‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (ℜ‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3946 ⟶wf 6535 ‘cfv 6539 Fincfn 8934 ℂcc 11103 ℝcr 11104 ℜcre 15039 Σcsu 15627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-oi 9500 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-n0 12468 df-z 12554 df-uz 12818 df-rp 12970 df-fz 13480 df-fzo 13623 df-seq 13962 df-exp 14023 df-hash 14286 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15427 df-sum 15628 |
This theorem is referenced by: (None) |
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