Metamath Proof Explorer


Theorem sge0lempt

Description: If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses sge0lempt.xph 𝑥 𝜑
sge0lempt.a ( 𝜑𝐴𝑉 )
sge0lempt.b ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
sge0lempt.c ( ( 𝜑𝑥𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) )
sge0lempt.le ( ( 𝜑𝑥𝐴 ) → 𝐵𝐶 )
Assertion sge0lempt ( 𝜑 → ( Σ^ ‘ ( 𝑥𝐴𝐵 ) ) ≤ ( Σ^ ‘ ( 𝑥𝐴𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 sge0lempt.xph 𝑥 𝜑
2 sge0lempt.a ( 𝜑𝐴𝑉 )
3 sge0lempt.b ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
4 sge0lempt.c ( ( 𝜑𝑥𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) )
5 sge0lempt.le ( ( 𝜑𝑥𝐴 ) → 𝐵𝐶 )
6 eqid ( 𝑥𝐴𝐵 ) = ( 𝑥𝐴𝐵 )
7 1 3 6 fmptdf ( 𝜑 → ( 𝑥𝐴𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) )
8 eqid ( 𝑥𝐴𝐶 ) = ( 𝑥𝐴𝐶 )
9 1 4 8 fmptdf ( 𝜑 → ( 𝑥𝐴𝐶 ) : 𝐴 ⟶ ( 0 [,] +∞ ) )
10 nfv 𝑥 𝑦𝐴
11 1 10 nfan 𝑥 ( 𝜑𝑦𝐴 )
12 nfcv 𝑥 𝑦
13 12 nfcsb1 𝑥 𝑦 / 𝑥 𝐵
14 nfcv 𝑥
15 12 nfcsb1 𝑥 𝑦 / 𝑥 𝐶
16 13 14 15 nfbr 𝑥 𝑦 / 𝑥 𝐵 𝑦 / 𝑥 𝐶
17 11 16 nfim 𝑥 ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐵 𝑦 / 𝑥 𝐶 )
18 eleq1w ( 𝑥 = 𝑦 → ( 𝑥𝐴𝑦𝐴 ) )
19 18 anbi2d ( 𝑥 = 𝑦 → ( ( 𝜑𝑥𝐴 ) ↔ ( 𝜑𝑦𝐴 ) ) )
20 csbeq1a ( 𝑥 = 𝑦𝐵 = 𝑦 / 𝑥 𝐵 )
21 csbeq1a ( 𝑥 = 𝑦𝐶 = 𝑦 / 𝑥 𝐶 )
22 20 21 breq12d ( 𝑥 = 𝑦 → ( 𝐵𝐶 𝑦 / 𝑥 𝐵 𝑦 / 𝑥 𝐶 ) )
23 19 22 imbi12d ( 𝑥 = 𝑦 → ( ( ( 𝜑𝑥𝐴 ) → 𝐵𝐶 ) ↔ ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐵 𝑦 / 𝑥 𝐶 ) ) )
24 17 23 5 chvarfv ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐵 𝑦 / 𝑥 𝐶 )
25 simpr ( ( 𝜑𝑦𝐴 ) → 𝑦𝐴 )
26 13 nfel1 𝑥 𝑦 / 𝑥 𝐵 ∈ ( 0 [,] +∞ )
27 11 26 nfim 𝑥 ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐵 ∈ ( 0 [,] +∞ ) )
28 20 eleq1d ( 𝑥 = 𝑦 → ( 𝐵 ∈ ( 0 [,] +∞ ) ↔ 𝑦 / 𝑥 𝐵 ∈ ( 0 [,] +∞ ) ) )
29 19 28 imbi12d ( 𝑥 = 𝑦 → ( ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) ↔ ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐵 ∈ ( 0 [,] +∞ ) ) ) )
30 27 29 3 chvarfv ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐵 ∈ ( 0 [,] +∞ ) )
31 12 13 20 6 fvmptf ( ( 𝑦𝐴 𝑦 / 𝑥 𝐵 ∈ ( 0 [,] +∞ ) ) → ( ( 𝑥𝐴𝐵 ) ‘ 𝑦 ) = 𝑦 / 𝑥 𝐵 )
32 25 30 31 syl2anc ( ( 𝜑𝑦𝐴 ) → ( ( 𝑥𝐴𝐵 ) ‘ 𝑦 ) = 𝑦 / 𝑥 𝐵 )
33 nfcv 𝑥 ( 0 [,] +∞ )
34 15 33 nfel 𝑥 𝑦 / 𝑥 𝐶 ∈ ( 0 [,] +∞ )
35 11 34 nfim 𝑥 ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐶 ∈ ( 0 [,] +∞ ) )
36 21 eleq1d ( 𝑥 = 𝑦 → ( 𝐶 ∈ ( 0 [,] +∞ ) ↔ 𝑦 / 𝑥 𝐶 ∈ ( 0 [,] +∞ ) ) )
37 19 36 imbi12d ( 𝑥 = 𝑦 → ( ( ( 𝜑𝑥𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) ↔ ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐶 ∈ ( 0 [,] +∞ ) ) ) )
38 35 37 4 chvarfv ( ( 𝜑𝑦𝐴 ) → 𝑦 / 𝑥 𝐶 ∈ ( 0 [,] +∞ ) )
39 12 15 21 8 fvmptf ( ( 𝑦𝐴 𝑦 / 𝑥 𝐶 ∈ ( 0 [,] +∞ ) ) → ( ( 𝑥𝐴𝐶 ) ‘ 𝑦 ) = 𝑦 / 𝑥 𝐶 )
40 25 38 39 syl2anc ( ( 𝜑𝑦𝐴 ) → ( ( 𝑥𝐴𝐶 ) ‘ 𝑦 ) = 𝑦 / 𝑥 𝐶 )
41 32 40 breq12d ( ( 𝜑𝑦𝐴 ) → ( ( ( 𝑥𝐴𝐵 ) ‘ 𝑦 ) ≤ ( ( 𝑥𝐴𝐶 ) ‘ 𝑦 ) ↔ 𝑦 / 𝑥 𝐵 𝑦 / 𝑥 𝐶 ) )
42 24 41 mpbird ( ( 𝜑𝑦𝐴 ) → ( ( 𝑥𝐴𝐵 ) ‘ 𝑦 ) ≤ ( ( 𝑥𝐴𝐶 ) ‘ 𝑦 ) )
43 2 7 9 42 sge0le ( 𝜑 → ( Σ^ ‘ ( 𝑥𝐴𝐵 ) ) ≤ ( Σ^ ‘ ( 𝑥𝐴𝐶 ) ) )