Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumsupp0 | Structured version Visualization version GIF version |
Description: Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
fsumsupp0.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumsupp0.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
Ref | Expression |
---|---|
fsumsupp0 | ⊢ (𝜑 → Σ𝑘 ∈ (𝐹 supp 0)(𝐹‘𝑘) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑘)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumsupp0.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
2 | 1 | ffnd 6505 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | fsumsupp0.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
4 | 0red 10722 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) | |
5 | suppvalfn 7864 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin ∧ 0 ∈ ℝ) → (𝐹 supp 0) = {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0}) | |
6 | 2, 3, 4, 5 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝐹 supp 0) = {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0}) |
7 | ssrab2 3969 | . . 3 ⊢ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0} ⊆ 𝐴 | |
8 | 6, 7 | eqsstrdi 3931 | . 2 ⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐴) |
9 | 1 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐹 supp 0)) → 𝐹:𝐴⟶ℂ) |
10 | 8 | sselda 3877 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐹 supp 0)) → 𝑘 ∈ 𝐴) |
11 | 9, 10 | ffvelrnd 6862 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐹 supp 0)) → (𝐹‘𝑘) ∈ ℂ) |
12 | eldifi 4017 | . . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) → 𝑘 ∈ 𝐴) | |
13 | 12 | adantr 484 | . . . . . . 7 ⊢ ((𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) ∧ ¬ (𝐹‘𝑘) = 0) → 𝑘 ∈ 𝐴) |
14 | neqne 2942 | . . . . . . . 8 ⊢ (¬ (𝐹‘𝑘) = 0 → (𝐹‘𝑘) ≠ 0) | |
15 | 14 | adantl 485 | . . . . . . 7 ⊢ ((𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) ∧ ¬ (𝐹‘𝑘) = 0) → (𝐹‘𝑘) ≠ 0) |
16 | 13, 15 | jca 515 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) ∧ ¬ (𝐹‘𝑘) = 0) → (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 0)) |
17 | rabid 3281 | . . . . . 6 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0} ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 0)) | |
18 | 16, 17 | sylibr 237 | . . . . 5 ⊢ ((𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) ∧ ¬ (𝐹‘𝑘) = 0) → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0}) |
19 | 18 | adantll 714 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) ∧ ¬ (𝐹‘𝑘) = 0) → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0}) |
20 | 6 | eleq2d 2818 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ (𝐹 supp 0) ↔ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0})) |
21 | 20 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) ∧ ¬ (𝐹‘𝑘) = 0) → (𝑘 ∈ (𝐹 supp 0) ↔ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0})) |
22 | 19, 21 | mpbird 260 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) ∧ ¬ (𝐹‘𝑘) = 0) → 𝑘 ∈ (𝐹 supp 0)) |
23 | eldifn 4018 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) → ¬ 𝑘 ∈ (𝐹 supp 0)) | |
24 | 23 | ad2antlr 727 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) ∧ ¬ (𝐹‘𝑘) = 0) → ¬ 𝑘 ∈ (𝐹 supp 0)) |
25 | 22, 24 | condan 818 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) → (𝐹‘𝑘) = 0) |
26 | 8, 11, 25, 3 | fsumss 15175 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝐹 supp 0)(𝐹‘𝑘) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑘)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 {crab 3057 ∖ cdif 3840 Fn wfn 6334 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 supp csupp 7856 Fincfn 8555 ℂcc 10613 ℝcr 10614 0cc0 10615 Σcsu 15135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-sum 15136 |
This theorem is referenced by: rrxtopnfi 43370 |
Copyright terms: Public domain | W3C validator |