Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0snmpt | Structured version Visualization version GIF version |
Description: A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0snmpt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0snmpt.c | ⊢ (𝜑 → 𝐶 ∈ (0[,]+∞)) |
sge0snmpt.b | ⊢ (𝑘 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sge0snmpt | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0snmpt.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | elsni 4579 | . . . . . . 7 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴) | |
3 | sge0snmpt.b | . . . . . . 7 ⊢ (𝑘 = 𝐴 → 𝐵 = 𝐶) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝑘 ∈ {𝐴} → 𝐵 = 𝐶) |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → 𝐵 = 𝐶) |
6 | sge0snmpt.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (0[,]+∞)) | |
7 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → 𝐶 ∈ (0[,]+∞)) |
8 | 5, 7 | eqeltrd 2839 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → 𝐵 ∈ (0[,]+∞)) |
9 | eqid 2738 | . . . 4 ⊢ (𝑘 ∈ {𝐴} ↦ 𝐵) = (𝑘 ∈ {𝐴} ↦ 𝐵) | |
10 | 8, 9 | fmptd 6981 | . . 3 ⊢ (𝜑 → (𝑘 ∈ {𝐴} ↦ 𝐵):{𝐴}⟶(0[,]+∞)) |
11 | 1, 10 | sge0sn 43876 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = ((𝑘 ∈ {𝐴} ↦ 𝐵)‘𝐴)) |
12 | eqidd 2739 | . . 3 ⊢ (𝜑 → (𝑘 ∈ {𝐴} ↦ 𝐵) = (𝑘 ∈ {𝐴} ↦ 𝐵)) | |
13 | 3 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐵 = 𝐶) |
14 | snidg 4596 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
15 | 1, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ {𝐴}) |
16 | 12, 13, 15, 6 | fvmptd 6875 | . 2 ⊢ (𝜑 → ((𝑘 ∈ {𝐴} ↦ 𝐵)‘𝐴) = 𝐶) |
17 | 11, 16 | eqtrd 2778 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {csn 4562 ↦ cmpt 5157 ‘cfv 6427 (class class class)co 7268 0cc0 10859 +∞cpnf 10994 [,]cicc 13070 Σ^csumge0 43859 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-inf2 9387 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-sup 9189 df-oi 9257 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-n0 12222 df-z 12308 df-uz 12571 df-rp 12719 df-ico 13073 df-icc 13074 df-fz 13228 df-fzo 13371 df-seq 13710 df-exp 13771 df-hash 14033 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-clim 15185 df-sum 15386 df-sumge0 43860 |
This theorem is referenced by: sge0prle 43898 sge0p1 43911 ovnhoilem1 44098 |
Copyright terms: Public domain | W3C validator |