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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 46350. (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 4738 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐵, 𝐵}) | |
2 | dfsn2 4644 | . . . . . . . . . 10 ⊢ {𝐵} = {𝐵, 𝐵} | |
3 | 2 | eqcomi 2744 | . . . . . . . . 9 ⊢ {𝐵, 𝐵} = {𝐵} |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐵} = {𝐵}) |
5 | 1, 4 | eqtrd 2775 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐵}) |
6 | 5 | mpteq1d 5243 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐵} ↦ 𝐶)) |
7 | 6 | fveq2d 6911 | . . . . 5 ⊢ (𝐴 = 𝐵 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶))) |
8 | 7 | adantl 481 | . . . 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 46339 | . . . . 5 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐸) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐸) |
14 | 8, 13 | eqtrd 2775 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = 𝐸) |
15 | iccssxr 13467 | . . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ* | |
16 | 15, 10 | sselid 3993 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ*) |
17 | 16 | xaddlidd 45270 | . . . . . 6 ⊢ (𝜑 → (0 +𝑒 𝐸) = 𝐸) |
18 | 17 | eqcomd 2741 | . . . . 5 ⊢ (𝜑 → 𝐸 = (0 +𝑒 𝐸)) |
19 | 0xr 11306 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
21 | sge0prle.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) | |
22 | 15, 21 | sselid 3993 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ*) |
23 | pnfxr 11313 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ*) |
25 | iccgelb 13440 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷) | |
26 | 20, 24, 21, 25 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐷) |
27 | 20, 22, 16, 26 | xleadd1d 45279 | . . . . 5 ⊢ (𝜑 → (0 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
28 | 18, 27 | eqbrtrd 5170 | . . . 4 ⊢ (𝜑 → 𝐸 ≤ (𝐷 +𝑒 𝐸)) |
29 | 28 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐸 ≤ (𝐷 +𝑒 𝐸)) |
30 | 14, 29 | eqbrtrd 5170 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
31 | sge0prle.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
32 | 31 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑉) |
33 | 9 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊) |
34 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
35 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐸 ∈ (0[,]+∞)) |
36 | sge0prle.cd | . . . 4 ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) | |
37 | neqne 2946 | . . . . 5 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | |
38 | 37 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵) |
39 | 32, 33, 34, 35, 36, 11, 38 | sge0pr 46350 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
40 | 22, 16 | xaddcld 13340 | . . . . 5 ⊢ (𝜑 → (𝐷 +𝑒 𝐸) ∈ ℝ*) |
41 | 40 | xrleidd 13191 | . . . 4 ⊢ (𝜑 → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
42 | 41 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐸) ≤ (𝐷 +𝑒 𝐸)) |
43 | 39, 42 | eqbrtrd 5170 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
44 | 30, 43 | pm2.61dan 813 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {csn 4631 {cpr 4633 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 ℝ*cxr 11292 ≤ cle 11294 +𝑒 cxad 13150 [,]cicc 13387 Σ^csumge0 46318 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-xadd 13153 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-sumge0 46319 |
This theorem is referenced by: omeunle 46472 |
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