Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > esummulc2 | Structured version Visualization version GIF version |
Description: An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
Ref | Expression |
---|---|
esummulc2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esummulc2.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esummulc2.c | ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
Ref | Expression |
---|---|
esummulc2 | ⊢ (𝜑 → (𝐶 ·e Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶 ·e 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icossxr 13152 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ* | |
2 | esummulc2.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) | |
3 | 1, 2 | sselid 3919 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
4 | iccssxr 13150 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
5 | esummulc2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | esummulc2.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
7 | 6 | ralrimiva 3113 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
8 | nfcv 2907 | . . . . . 6 ⊢ Ⅎ𝑘𝐴 | |
9 | 8 | esumcl 31984 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
10 | 5, 7, 9 | syl2anc 584 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
11 | 4, 10 | sselid 3919 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
12 | xmulcom 12988 | . . 3 ⊢ ((𝐶 ∈ ℝ* ∧ Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) → (𝐶 ·e Σ*𝑘 ∈ 𝐴𝐵) = (Σ*𝑘 ∈ 𝐴𝐵 ·e 𝐶)) | |
13 | 3, 11, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐶 ·e Σ*𝑘 ∈ 𝐴𝐵) = (Σ*𝑘 ∈ 𝐴𝐵 ·e 𝐶)) |
14 | 5, 6, 2 | esummulc1 32035 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 ·e 𝐶) = Σ*𝑘 ∈ 𝐴(𝐵 ·e 𝐶)) |
15 | 4, 6 | sselid 3919 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
16 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ*) |
17 | xmulcom 12988 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ·e 𝐶) = (𝐶 ·e 𝐵)) | |
18 | 15, 16, 17 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 ·e 𝐶) = (𝐶 ·e 𝐵)) |
19 | 18 | esumeq2dv 31992 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐵 ·e 𝐶) = Σ*𝑘 ∈ 𝐴(𝐶 ·e 𝐵)) |
20 | 13, 14, 19 | 3eqtrd 2782 | 1 ⊢ (𝜑 → (𝐶 ·e Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶 ·e 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 (class class class)co 7268 0cc0 10859 +∞cpnf 10994 ℝ*cxr 10996 ·e cxmu 12835 [,)cico 13069 [,]cicc 13070 Σ*cesum 31981 |
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-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-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 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-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 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-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-q 12677 df-rp 12719 df-xneg 12836 df-xadd 12837 df-xmul 12838 df-ioo 13071 df-ioc 13072 df-ico 13073 df-icc 13074 df-fz 13228 df-fzo 13371 df-seq 13710 df-hash 14033 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-tset 16969 df-ple 16970 df-ds 16972 df-rest 17121 df-topn 17122 df-0g 17140 df-gsum 17141 df-topgen 17142 df-ordt 17200 df-xrs 17201 df-mre 17283 df-mrc 17284 df-acs 17286 df-ps 18272 df-tsr 18273 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-mhm 18418 df-submnd 18419 df-cntz 18911 df-cmn 19376 df-fbas 20582 df-fg 20583 df-top 22031 df-topon 22048 df-topsp 22070 df-bases 22084 df-ntr 22159 df-nei 22237 df-cn 22366 df-cnp 22367 df-haus 22454 df-fil 22985 df-fm 23077 df-flim 23078 df-flf 23079 df-tsms 23266 df-esum 31982 |
This theorem is referenced by: omssubadd 32253 |
Copyright terms: Public domain | W3C validator |