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Mathbox for Glauco Siliprandi |
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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 6718 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | fsumsupp0.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
4 | 0red 11224 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) | |
5 | suppvalfn 8159 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin ∧ 0 ∈ ℝ) → (𝐹 supp 0) = {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0}) | |
6 | 2, 3, 4, 5 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝐹 supp 0) = {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0}) |
7 | ssrab2 4077 | . . 3 ⊢ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0} ⊆ 𝐴 | |
8 | 6, 7 | eqsstrdi 4036 | . 2 ⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐴) |
9 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐹 supp 0)) → 𝐹:𝐴⟶ℂ) |
10 | 8 | sselda 3982 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐹 supp 0)) → 𝑘 ∈ 𝐴) |
11 | 9, 10 | ffvelcdmd 7087 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐹 supp 0)) → (𝐹‘𝑘) ∈ ℂ) |
12 | eldifi 4126 | . . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) → 𝑘 ∈ 𝐴) | |
13 | 12 | adantr 480 | . . . . . . 7 ⊢ ((𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) ∧ ¬ (𝐹‘𝑘) = 0) → 𝑘 ∈ 𝐴) |
14 | neqne 2947 | . . . . . . . 8 ⊢ (¬ (𝐹‘𝑘) = 0 → (𝐹‘𝑘) ≠ 0) | |
15 | 14 | adantl 481 | . . . . . . 7 ⊢ ((𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) ∧ ¬ (𝐹‘𝑘) = 0) → (𝐹‘𝑘) ≠ 0) |
16 | 13, 15 | jca 511 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) ∧ ¬ (𝐹‘𝑘) = 0) → (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 0)) |
17 | rabid 3451 | . . . . . 6 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0} ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 0)) | |
18 | 16, 17 | sylibr 233 | . . . . 5 ⊢ ((𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) ∧ ¬ (𝐹‘𝑘) = 0) → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0}) |
19 | 18 | adantll 711 | . . . 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 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) ∧ ¬ (𝐹‘𝑘) = 0) → (𝑘 ∈ (𝐹 supp 0) ↔ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ (𝐹‘𝑘) ≠ 0})) |
22 | 19, 21 | mpbird 257 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) ∧ ¬ (𝐹‘𝑘) = 0) → 𝑘 ∈ (𝐹 supp 0)) |
23 | eldifn 4127 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ (𝐹 supp 0)) → ¬ 𝑘 ∈ (𝐹 supp 0)) | |
24 | 23 | ad2antlr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) ∧ ¬ (𝐹‘𝑘) = 0) → ¬ 𝑘 ∈ (𝐹 supp 0)) |
25 | 22, 24 | condan 815 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 0))) → (𝐹‘𝑘) = 0) |
26 | 8, 11, 25, 3 | fsumss 15678 | 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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 {crab 3431 ∖ cdif 3945 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 supp csupp 8151 Fincfn 8945 ℂcc 11114 ℝcr 11115 0cc0 11116 Σcsu 15639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 |
This theorem is referenced by: rrxtopnfi 45462 |
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