fsumlessf - Mathbox for Glauco Siliprandi

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Theorem fsumlessf 45198

Description: A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

**Hypotheses**
Ref	Expression
fsumlessf.k	⊢ Ⅎ𝑘𝜑
fsumge0.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumge0.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
fsumge0.l	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)
fsumless.c	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

**Assertion**
Ref	Expression
fsumlessf	⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵)

Distinct variable groups: 𝐴,𝑘 𝐶,𝑘 Allowed substitution hints: 𝜑(𝑘) 𝐵(𝑘)

Proof of Theorem fsumlessf Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsumge0.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		fsumlessf.k	. . . . . 6 ⊢ Ⅎ𝑘𝜑
3		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝐴
4	2, 3	nfan 1895	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝐴)
5		nfcsb1v 3917	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
6	5	nfel1 2909	. . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ
7	4, 6	nfim 1892	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
8		eleq1w 2809	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴))
9	8	anbi2d 628	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑗 ∈ 𝐴)))
10		csbeq1a 3906	. . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
11	10	eleq1d 2811	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ))
12	9, 11	imbi12d 343	. . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)))
13		fsumge0.b	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
14	7, 12, 13	chvarfv 2229	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℝ)
15		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑘0
16		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑘 ≤
17	15, 16, 5	nfbr 5200	. . . . 5 ⊢ Ⅎ𝑘0 ≤ ⦋𝑗 / 𝑘⦌𝐵
18	4, 17	nfim 1892	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝐴) → 0 ≤ ⦋𝑗 / 𝑘⦌𝐵)
19	10	breq2d 5165	. . . . 5 ⊢ (𝑘 = 𝑗 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑗 / 𝑘⦌𝐵))
20	9, 19	imbi12d 343	. . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 0 ≤ ⦋𝑗 / 𝑘⦌𝐵)))
21		fsumge0.l	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)
22	18, 20, 21	chvarfv 2229	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 0 ≤ ⦋𝑗 / 𝑘⦌𝐵)
23		fsumless.c	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
24	1, 14, 22, 23	fsumless 15800	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝐶 ⦋𝑗 / 𝑘⦌𝐵 ≤ Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵)
25		nfcv 2892	. . . 4 ⊢ Ⅎ𝑗𝐶
26		nfcv 2892	. . . 4 ⊢ Ⅎ𝑘𝐶
27		nfcv 2892	. . . 4 ⊢ Ⅎ𝑗𝐵
28	10, 25, 26, 27, 5	cbvsum 15699	. . 3 ⊢ Σ𝑘 ∈ 𝐶 𝐵 = Σ𝑗 ∈ 𝐶 ⦋𝑗 / 𝑘⦌𝐵
29		nfcv 2892	. . . 4 ⊢ Ⅎ𝑗𝐴
30		nfcv 2892	. . . 4 ⊢ Ⅎ𝑘𝐴
31	10, 29, 30, 27, 5	cbvsum 15699	. . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵
32	28, 31	breq12i 5162	. 2 ⊢ (Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵 ↔ Σ𝑗 ∈ 𝐶 ⦋𝑗 / 𝑘⦌𝐵 ≤ Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵)
33	24, 32	sylibr 233	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ⦋csb 3892 ⊆ wss 3947 class class class wbr 5153 Fincfn 8974 ℝcr 11157 0cc0 11158 ≤ cle 11299 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-ico 13384 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: sge0uzfsumgt 46065 sge0reuz 46068

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