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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0rnre | Structured version Visualization version GIF version |
Description: When Σ^ is applied to nonnegative real numbers the range used in its definition is a subset of the reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0rnre.1 | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) |
Ref | Expression |
---|---|
sge0rnre | ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel2 4187 | . . . . 5 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin) | |
2 | 1 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin) |
3 | rge0ssre 13410 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
4 | sge0rnre.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) | |
5 | 4 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,)+∞)) |
6 | elinel1 4186 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋) | |
7 | elpwi 4598 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
8 | 6, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋) |
9 | 8 | adantr 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝑋) |
10 | simpr 485 | . . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥) | |
11 | 9, 10 | sseldd 3974 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋) |
12 | 11 | adantll 712 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋) |
13 | 5, 12 | ffvelcdmd 7067 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞)) |
14 | 3, 13 | sselid 3971 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ) |
15 | 2, 14 | fsumrecl 15657 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ) |
16 | 15 | ralrimiva 3145 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ) |
17 | eqid 2731 | . . 3 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) | |
18 | 17 | rnmptss 7101 | . 2 ⊢ (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ) |
19 | 16, 18 | syl 17 | 1 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3060 ∩ cin 3938 ⊆ wss 3939 𝒫 cpw 4591 ↦ cmpt 5219 ran crn 5665 ⟶wf 6523 ‘cfv 6527 (class class class)co 7388 Fincfn 8917 ℝcr 11086 0cc0 11087 +∞cpnf 11222 [,)cico 13303 Σcsu 15609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5273 ax-sep 5287 ax-nul 5294 ax-pow 5351 ax-pr 5415 ax-un 7703 ax-inf2 9613 ax-cnex 11143 ax-resscn 11144 ax-1cn 11145 ax-icn 11146 ax-addcl 11147 ax-addrcl 11148 ax-mulcl 11149 ax-mulrcl 11150 ax-mulcom 11151 ax-addass 11152 ax-mulass 11153 ax-distr 11154 ax-i2m1 11155 ax-1ne0 11156 ax-1rid 11157 ax-rnegex 11158 ax-rrecex 11159 ax-cnre 11160 ax-pre-lttri 11161 ax-pre-lttrn 11162 ax-pre-ltadd 11163 ax-pre-mulgt0 11164 ax-pre-sup 11165 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3374 df-reu 3375 df-rab 3429 df-v 3471 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4314 df-if 4518 df-pw 4593 df-sn 4618 df-pr 4620 df-op 4624 df-uni 4897 df-int 4939 df-iun 4987 df-br 5137 df-opab 5199 df-mpt 5220 df-tr 5254 df-id 5562 df-eprel 5568 df-po 5576 df-so 5577 df-fr 5619 df-se 5620 df-we 5621 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6284 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7344 df-ov 7391 df-oprab 7392 df-mpo 7393 df-om 7834 df-1st 7952 df-2nd 7953 df-frecs 8243 df-wrecs 8274 df-recs 8348 df-rdg 8387 df-1o 8443 df-er 8681 df-en 8918 df-dom 8919 df-sdom 8920 df-fin 8921 df-sup 9414 df-oi 9482 df-card 9911 df-pnf 11227 df-mnf 11228 df-xr 11229 df-ltxr 11230 df-le 11231 df-sub 11423 df-neg 11424 df-div 11849 df-nn 12190 df-2 12252 df-3 12253 df-n0 12450 df-z 12536 df-uz 12800 df-rp 12952 df-ico 13307 df-fz 13462 df-fzo 13605 df-seq 13944 df-exp 14005 df-hash 14268 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-clim 15409 df-sum 15610 |
This theorem is referenced by: fsumlesge0 44803 sge0supre 44815 sge0less 44818 sge0ltfirp 44826 sge0resplit 44832 sge0split 44835 |
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