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Mathbox for Glauco Siliprandi |
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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 41114. (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 4412 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐵, 𝐵}) | |
2 | dfsn2 4334 | . . . . . . . . . 10 ⊢ {𝐵} = {𝐵, 𝐵} | |
3 | 2 | eqcomi 2769 | . . . . . . . . 9 ⊢ {𝐵, 𝐵} = {𝐵} |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐵} = {𝐵}) |
5 | 1, 4 | eqtrd 2794 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐵}) |
6 | 5 | mpteq1d 4890 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐵} ↦ 𝐶)) |
7 | 6 | fveq2d 6356 | . . . . 5 ⊢ (𝐴 = 𝐵 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶))) |
8 | 7 | adantl 473 | . . . 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 41103 | . . . . 5 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐸) |
13 | 12 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐸) |
14 | 8, 13 | eqtrd 2794 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = 𝐸) |
15 | iccssxr 12449 | . . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ* | |
16 | 15, 10 | sseldi 3742 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ*) |
17 | 16 | xaddid2d 40033 | . . . . . 6 ⊢ (𝜑 → (0 +𝑒 𝐸) = 𝐸) |
18 | 17 | eqcomd 2766 | . . . . 5 ⊢ (𝜑 → 𝐸 = (0 +𝑒 𝐸)) |
19 | 0xr 10278 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
21 | sge0prle.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) | |
22 | 15, 21 | sseldi 3742 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ*) |
23 | pnfxr 10284 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ*) |
25 | iccgelb 12423 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷) | |
26 | 20, 24, 21, 25 | syl3anc 1477 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐷) |
27 | 20, 22, 16, 26 | xleadd1d 40043 | . . . . 5 ⊢ (𝜑 → (0 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
28 | 18, 27 | eqbrtrd 4826 | . . . 4 ⊢ (𝜑 → 𝐸 ≤ (𝐷 +𝑒 𝐸)) |
29 | 28 | adantr 472 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐸 ≤ (𝐷 +𝑒 𝐸)) |
30 | 14, 29 | eqbrtrd 4826 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
31 | sge0prle.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
32 | 31 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉) |
33 | 9 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊) |
34 | 21 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
35 | 10 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐸 ∈ (0[,]+∞)) |
36 | sge0prle.cd | . . . 4 ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) | |
37 | neqne 2940 | . . . . 5 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | |
38 | 37 | adantl 473 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵) |
39 | 32, 33, 34, 35, 36, 11, 38 | sge0pr 41114 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
40 | 22, 16 | xaddcld 12324 | . . . . 5 ⊢ (𝜑 → (𝐷 +𝑒 𝐸) ∈ ℝ*) |
41 | xrleid 12176 | . . . . 5 ⊢ ((𝐷 +𝑒 𝐸) ∈ ℝ* → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) | |
42 | 40, 41 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
43 | 42 | adantr 472 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
44 | 39, 43 | eqbrtrd 4826 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
45 | 30, 44 | pm2.61dan 867 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 {csn 4321 {cpr 4323 class class class wbr 4804 ↦ cmpt 4881 ‘cfv 6049 (class class class)co 6813 0cc0 10128 +∞cpnf 10263 ℝ*cxr 10265 ≤ cle 10267 +𝑒 cxad 12137 [,]cicc 12371 Σ^csumge0 41082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-oi 8580 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-xadd 12140 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-sum 14616 df-sumge0 41083 |
This theorem is referenced by: omeunle 41236 |
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