Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0prle | Structured version Visualization version GIF version |
Description: The sum of a pair of nonnegative extended reals is less than or equal their extended addition. When it is a distinct pair, than equality holds, see sge0pr 42669. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0prle.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0prle.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
sge0prle.d | ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) |
sge0prle.e | ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞)) |
sge0prle.cd | ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) |
sge0prle.ce | ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
sge0prle | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 4663 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐵, 𝐵}) | |
2 | dfsn2 4574 | . . . . . . . . . 10 ⊢ {𝐵} = {𝐵, 𝐵} | |
3 | 2 | eqcomi 2830 | . . . . . . . . 9 ⊢ {𝐵, 𝐵} = {𝐵} |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐵} = {𝐵}) |
5 | 1, 4 | eqtrd 2856 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐵}) |
6 | 5 | mpteq1d 5148 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐵} ↦ 𝐶)) |
7 | 6 | fveq2d 6669 | . . . . 5 ⊢ (𝐴 = 𝐵 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶))) |
8 | 7 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶))) |
9 | sge0prle.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
10 | sge0prle.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞)) | |
11 | sge0prle.ce | . . . . . 6 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸) | |
12 | 9, 10, 11 | sge0snmpt 42658 | . . . . 5 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐸) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐸) |
14 | 8, 13 | eqtrd 2856 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = 𝐸) |
15 | iccssxr 12813 | . . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ* | |
16 | 15, 10 | sseldi 3965 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ*) |
17 | 16 | xaddid2d 41579 | . . . . . 6 ⊢ (𝜑 → (0 +𝑒 𝐸) = 𝐸) |
18 | 17 | eqcomd 2827 | . . . . 5 ⊢ (𝜑 → 𝐸 = (0 +𝑒 𝐸)) |
19 | 0xr 10682 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
21 | sge0prle.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) | |
22 | 15, 21 | sseldi 3965 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ*) |
23 | pnfxr 10689 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ*) |
25 | iccgelb 12787 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷) | |
26 | 20, 24, 21, 25 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐷) |
27 | 20, 22, 16, 26 | xleadd1d 41589 | . . . . 5 ⊢ (𝜑 → (0 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
28 | 18, 27 | eqbrtrd 5081 | . . . 4 ⊢ (𝜑 → 𝐸 ≤ (𝐷 +𝑒 𝐸)) |
29 | 28 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐸 ≤ (𝐷 +𝑒 𝐸)) |
30 | 14, 29 | eqbrtrd 5081 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
31 | sge0prle.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
32 | 31 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉) |
33 | 9 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊) |
34 | 21 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
35 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐸 ∈ (0[,]+∞)) |
36 | sge0prle.cd | . . . 4 ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) | |
37 | neqne 3024 | . . . . 5 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | |
38 | 37 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵) |
39 | 32, 33, 34, 35, 36, 11, 38 | sge0pr 42669 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
40 | 22, 16 | xaddcld 12688 | . . . . 5 ⊢ (𝜑 → (𝐷 +𝑒 𝐸) ∈ ℝ*) |
41 | 40 | xrleidd 12539 | . . . 4 ⊢ (𝜑 → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
42 | 41 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
43 | 39, 42 | eqbrtrd 5081 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
44 | 30, 43 | pm2.61dan 811 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {csn 4561 {cpr 4563 class class class wbr 5059 ↦ cmpt 5139 ‘cfv 6350 (class class class)co 7150 0cc0 10531 +∞cpnf 10666 ℝ*cxr 10668 ≤ cle 10670 +𝑒 cxad 12499 [,]cicc 12735 Σ^csumge0 42637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-xadd 12502 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-sumge0 42638 |
This theorem is referenced by: omeunle 42791 |
Copyright terms: Public domain | W3C validator |