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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 45829. (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 4742 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐵, 𝐵}) | |
2 | dfsn2 4645 | . . . . . . . . . 10 ⊢ {𝐵} = {𝐵, 𝐵} | |
3 | 2 | eqcomi 2737 | . . . . . . . . 9 ⊢ {𝐵, 𝐵} = {𝐵} |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐵} = {𝐵}) |
5 | 1, 4 | eqtrd 2768 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐵}) |
6 | 5 | mpteq1d 5247 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐵} ↦ 𝐶)) |
7 | 6 | fveq2d 6906 | . . . . 5 ⊢ (𝐴 = 𝐵 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶))) |
8 | 7 | adantl 480 | . . . 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 45818 | . . . . 5 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐸) |
13 | 12 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐸) |
14 | 8, 13 | eqtrd 2768 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = 𝐸) |
15 | iccssxr 13449 | . . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ* | |
16 | 15, 10 | sselid 3980 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ*) |
17 | 16 | xaddlidd 44748 | . . . . . 6 ⊢ (𝜑 → (0 +𝑒 𝐸) = 𝐸) |
18 | 17 | eqcomd 2734 | . . . . 5 ⊢ (𝜑 → 𝐸 = (0 +𝑒 𝐸)) |
19 | 0xr 11301 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
21 | sge0prle.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) | |
22 | 15, 21 | sselid 3980 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ*) |
23 | pnfxr 11308 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ*) |
25 | iccgelb 13422 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷) | |
26 | 20, 24, 21, 25 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐷) |
27 | 20, 22, 16, 26 | xleadd1d 44758 | . . . . 5 ⊢ (𝜑 → (0 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
28 | 18, 27 | eqbrtrd 5174 | . . . 4 ⊢ (𝜑 → 𝐸 ≤ (𝐷 +𝑒 𝐸)) |
29 | 28 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐸 ≤ (𝐷 +𝑒 𝐸)) |
30 | 14, 29 | eqbrtrd 5174 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
31 | sge0prle.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
32 | 31 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉) |
33 | 9 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊) |
34 | 21 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
35 | 10 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐸 ∈ (0[,]+∞)) |
36 | sge0prle.cd | . . . 4 ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) | |
37 | neqne 2945 | . . . . 5 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | |
38 | 37 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵) |
39 | 32, 33, 34, 35, 36, 11, 38 | sge0pr 45829 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
40 | 22, 16 | xaddcld 13322 | . . . . 5 ⊢ (𝜑 → (𝐷 +𝑒 𝐸) ∈ ℝ*) |
41 | 40 | xrleidd 13173 | . . . 4 ⊢ (𝜑 → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
42 | 41 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
43 | 39, 42 | eqbrtrd 5174 | . 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 {csn 4632 {cpr 4634 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 0cc0 11148 +∞cpnf 11285 ℝ*cxr 11287 ≤ cle 11289 +𝑒 cxad 13132 [,]cicc 13369 Σ^csumge0 45797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-xadd 13135 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 df-sum 15675 df-sumge0 45798 |
This theorem is referenced by: omeunle 45951 |
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