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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0rnn0 | Structured version Visualization version GIF version |
Description: The range used in the definition of Σ^ is not empty. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0rnn0 | ⊢ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 5360 | . . . . 5 ⊢ ∅ ∈ 𝒫 𝑋 | |
2 | 0fi 9080 | . . . . 5 ⊢ ∅ ∈ Fin | |
3 | 1, 2 | elini 4194 | . . . 4 ⊢ ∅ ∈ (𝒫 𝑋 ∩ Fin) |
4 | sum0 15725 | . . . . 5 ⊢ Σ𝑦 ∈ ∅ (𝐹‘𝑦) = 0 | |
5 | 4 | eqcomi 2735 | . . . 4 ⊢ 0 = Σ𝑦 ∈ ∅ (𝐹‘𝑦) |
6 | sumeq1 15693 | . . . . 5 ⊢ (𝑥 = ∅ → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = Σ𝑦 ∈ ∅ (𝐹‘𝑦)) | |
7 | 6 | rspceeqv 3630 | . . . 4 ⊢ ((∅ ∈ (𝒫 𝑋 ∩ Fin) ∧ 0 = Σ𝑦 ∈ ∅ (𝐹‘𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)0 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) |
8 | 3, 5, 7 | mp2an 690 | . . 3 ⊢ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)0 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) |
9 | 0re 11266 | . . . 4 ⊢ 0 ∈ ℝ | |
10 | eqid 2726 | . . . . 5 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) | |
11 | 10 | elrnmpt 5962 | . . . 4 ⊢ (0 ∈ ℝ → (0 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)0 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) |
12 | 9, 11 | ax-mp 5 | . . 3 ⊢ (0 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)0 = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) |
13 | 8, 12 | mpbir 230 | . 2 ⊢ 0 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) |
14 | ne0i 4337 | . 2 ⊢ (0 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ≠ ∅) | |
15 | 13, 14 | ax-mp 5 | 1 ⊢ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 ∩ cin 3946 ∅c0 4325 𝒫 cpw 4607 ↦ cmpt 5236 ran crn 5683 ‘cfv 6554 Fincfn 8974 ℝcr 11157 0cc0 11158 Σcsu 15690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-sum 15691 |
This theorem is referenced by: sge0supre 46010 sge0ltfirp 46021 sge0resplit 46027 |
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