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Theorem sge0pr 42553
Description: Sum of a pair of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0pr.a (𝜑𝐴𝑉)
sge0pr.b (𝜑𝐵𝑊)
sge0pr.d (𝜑𝐷 ∈ (0[,]+∞))
sge0pr.e (𝜑𝐸 ∈ (0[,]+∞))
sge0pr.cd (𝑘 = 𝐴𝐶 = 𝐷)
sge0pr.ce (𝑘 = 𝐵𝐶 = 𝐸)
sge0pr.ab (𝜑𝐴𝐵)
Assertion
Ref Expression
sge0pr (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝐷,𝑘   𝑘,𝐸   𝑘,𝑉   𝑘,𝑊   𝜑,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem sge0pr
StepHypRef Expression
1 iccssxr 12807 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
2 sge0pr.e . . . . . . 7 (𝜑𝐸 ∈ (0[,]+∞))
31, 2sseldi 3962 . . . . . 6 (𝜑𝐸 ∈ ℝ*)
4 mnfxr 10686 . . . . . . . 8 -∞ ∈ ℝ*
54a1i 11 . . . . . . 7 (𝜑 → -∞ ∈ ℝ*)
6 0xr 10676 . . . . . . . . 9 0 ∈ ℝ*
76a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℝ*)
8 mnflt0 12508 . . . . . . . . 9 -∞ < 0
98a1i 11 . . . . . . . 8 (𝜑 → -∞ < 0)
10 pnfxr 10683 . . . . . . . . . 10 +∞ ∈ ℝ*
1110a1i 11 . . . . . . . . 9 (𝜑 → +∞ ∈ ℝ*)
12 iccgelb 12781 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝐸 ∈ (0[,]+∞)) → 0 ≤ 𝐸)
137, 11, 2, 12syl3anc 1363 . . . . . . . 8 (𝜑 → 0 ≤ 𝐸)
145, 7, 3, 9, 13xrltletrd 12542 . . . . . . 7 (𝜑 → -∞ < 𝐸)
155, 3, 14xrgtned 41466 . . . . . 6 (𝜑𝐸 ≠ -∞)
16 xaddpnf2 12608 . . . . . 6 ((𝐸 ∈ ℝ*𝐸 ≠ -∞) → (+∞ +𝑒 𝐸) = +∞)
173, 15, 16syl2anc 584 . . . . 5 (𝜑 → (+∞ +𝑒 𝐸) = +∞)
1817eqcomd 2824 . . . 4 (𝜑 → +∞ = (+∞ +𝑒 𝐸))
1918adantr 481 . . 3 ((𝜑𝐷 = +∞) → +∞ = (+∞ +𝑒 𝐸))
20 prex 5323 . . . . 5 {𝐴, 𝐵} ∈ V
2120a1i 11 . . . 4 ((𝜑𝐷 = +∞) → {𝐴, 𝐵} ∈ V)
22 sge0pr.cd . . . . . . . . . 10 (𝑘 = 𝐴𝐶 = 𝐷)
2322adantl 482 . . . . . . . . 9 ((𝜑𝑘 = 𝐴) → 𝐶 = 𝐷)
24 sge0pr.d . . . . . . . . . 10 (𝜑𝐷 ∈ (0[,]+∞))
2524adantr 481 . . . . . . . . 9 ((𝜑𝑘 = 𝐴) → 𝐷 ∈ (0[,]+∞))
2623, 25eqeltrd 2910 . . . . . . . 8 ((𝜑𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
2726adantlr 711 . . . . . . 7 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
28 simpll 763 . . . . . . . 8 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
29 simpl 483 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ∈ {𝐴, 𝐵})
30 neqne 3021 . . . . . . . . . . 11 𝑘 = 𝐴𝑘𝐴)
3130adantl 482 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘𝐴)
32 elprn1 41790 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ 𝑘𝐴) → 𝑘 = 𝐵)
3329, 31, 32syl2anc 584 . . . . . . . . 9 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
3433adantll 710 . . . . . . . 8 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
35 sge0pr.ce . . . . . . . . . 10 (𝑘 = 𝐵𝐶 = 𝐸)
3635adantl 482 . . . . . . . . 9 ((𝜑𝑘 = 𝐵) → 𝐶 = 𝐸)
372adantr 481 . . . . . . . . 9 ((𝜑𝑘 = 𝐵) → 𝐸 ∈ (0[,]+∞))
3836, 37eqeltrd 2910 . . . . . . . 8 ((𝜑𝑘 = 𝐵) → 𝐶 ∈ (0[,]+∞))
3928, 34, 38syl2anc 584 . . . . . . 7 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
4027, 39pm2.61dan 809 . . . . . 6 ((𝜑𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,]+∞))
41 eqid 2818 . . . . . 6 (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)
4240, 41fmptd 6870 . . . . 5 (𝜑 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
4342adantr 481 . . . 4 ((𝜑𝐷 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
44 id 22 . . . . . . 7 (𝐷 = +∞ → 𝐷 = +∞)
4544eqcomd 2824 . . . . . 6 (𝐷 = +∞ → +∞ = 𝐷)
4645adantl 482 . . . . 5 ((𝜑𝐷 = +∞) → +∞ = 𝐷)
47 prid1g 4688 . . . . . . . 8 (𝐷 ∈ (0[,]+∞) → 𝐷 ∈ {𝐷, 𝐸})
4824, 47syl 17 . . . . . . 7 (𝜑𝐷 ∈ {𝐷, 𝐸})
49 sge0pr.a . . . . . . . . 9 (𝜑𝐴𝑉)
50 sge0pr.b . . . . . . . . 9 (𝜑𝐵𝑊)
5149, 50, 41, 22, 35rnmptpr 41309 . . . . . . . 8 (𝜑 → ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = {𝐷, 𝐸})
5251eqcomd 2824 . . . . . . 7 (𝜑 → {𝐷, 𝐸} = ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5348, 52eleqtrd 2912 . . . . . 6 (𝜑𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5453adantr 481 . . . . 5 ((𝜑𝐷 = +∞) → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5546, 54eqeltrd 2910 . . . 4 ((𝜑𝐷 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5621, 43, 55sge0pnfval 42532 . . 3 ((𝜑𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
57 oveq1 7152 . . . 4 (𝐷 = +∞ → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
5857adantl 482 . . 3 ((𝜑𝐷 = +∞) → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
5919, 56, 583eqtr4d 2863 . 2 ((𝜑𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
601, 24sseldi 3962 . . . . . . . 8 (𝜑𝐷 ∈ ℝ*)
61 iccgelb 12781 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷)
627, 11, 24, 61syl3anc 1363 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝐷)
635, 7, 60, 9, 62xrltletrd 12542 . . . . . . . . 9 (𝜑 → -∞ < 𝐷)
645, 60, 63xrgtned 41466 . . . . . . . 8 (𝜑𝐷 ≠ -∞)
65 xaddpnf1 12607 . . . . . . . 8 ((𝐷 ∈ ℝ*𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞)
6660, 64, 65syl2anc 584 . . . . . . 7 (𝜑 → (𝐷 +𝑒 +∞) = +∞)
6766eqcomd 2824 . . . . . 6 (𝜑 → +∞ = (𝐷 +𝑒 +∞))
6867adantr 481 . . . . 5 ((𝜑𝐸 = +∞) → +∞ = (𝐷 +𝑒 +∞))
6920a1i 11 . . . . . 6 ((𝜑𝐸 = +∞) → {𝐴, 𝐵} ∈ V)
7042adantr 481 . . . . . 6 ((𝜑𝐸 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
71 id 22 . . . . . . . . 9 (𝐸 = +∞ → 𝐸 = +∞)
7271eqcomd 2824 . . . . . . . 8 (𝐸 = +∞ → +∞ = 𝐸)
7372adantl 482 . . . . . . 7 ((𝜑𝐸 = +∞) → +∞ = 𝐸)
74 prid2g 4689 . . . . . . . . . 10 (𝐸 ∈ (0[,]+∞) → 𝐸 ∈ {𝐷, 𝐸})
752, 74syl 17 . . . . . . . . 9 (𝜑𝐸 ∈ {𝐷, 𝐸})
7675, 52eleqtrd 2912 . . . . . . . 8 (𝜑𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7776adantr 481 . . . . . . 7 ((𝜑𝐸 = +∞) → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7873, 77eqeltrd 2910 . . . . . 6 ((𝜑𝐸 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7969, 70, 78sge0pnfval 42532 . . . . 5 ((𝜑𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
80 oveq2 7153 . . . . . 6 (𝐸 = +∞ → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
8180adantl 482 . . . . 5 ((𝜑𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
8268, 79, 813eqtr4d 2863 . . . 4 ((𝜑𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
8382adantlr 711 . . 3 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
84 rge0ssre 12832 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
85 ax-resscn 10582 . . . . . . . 8 ℝ ⊆ ℂ
8684, 85sstri 3973 . . . . . . 7 (0[,)+∞) ⊆ ℂ
876a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ∈ ℝ*)
8810a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → +∞ ∈ ℝ*)
8960adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ ℝ*)
9062adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ≤ 𝐷)
91 pnfge 12513 . . . . . . . . . . . 12 (𝐷 ∈ ℝ*𝐷 ≤ +∞)
9260, 91syl 17 . . . . . . . . . . 11 (𝜑𝐷 ≤ +∞)
9392adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≤ +∞)
9444necon3bi 3039 . . . . . . . . . . 11 𝐷 = +∞ → 𝐷 ≠ +∞)
9594adantl 482 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≠ +∞)
9689, 88, 93, 95xrleneltd 41467 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 < +∞)
9787, 88, 89, 90, 96elicod 12775 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ (0[,)+∞))
9897adantr 481 . . . . . . 7 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ (0[,)+∞))
9986, 98sseldi 3962 . . . . . 6 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℂ)
1006a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ∈ ℝ*)
10110a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → +∞ ∈ ℝ*)
1023adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ*)
10313adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ≤ 𝐸)
104 pnfge 12513 . . . . . . . . . . . 12 (𝐸 ∈ ℝ*𝐸 ≤ +∞)
1053, 104syl 17 . . . . . . . . . . 11 (𝜑𝐸 ≤ +∞)
106105adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≤ +∞)
10771necon3bi 3039 . . . . . . . . . . 11 𝐸 = +∞ → 𝐸 ≠ +∞)
108107adantl 482 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≠ +∞)
109102, 101, 106, 108xrleneltd 41467 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 < +∞)
110100, 101, 102, 103, 109elicod 12775 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ (0[,)+∞))
11186, 110sseldi 3962 . . . . . . 7 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
112111adantlr 711 . . . . . 6 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
11399, 112jca 512 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
11449, 50jca 512 . . . . . 6 (𝜑 → (𝐴𝑉𝐵𝑊))
115114ad2antrr 722 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐴𝑉𝐵𝑊))
116 sge0pr.ab . . . . . 6 (𝜑𝐴𝐵)
117116ad2antrr 722 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐴𝐵)
11822, 35, 113, 115, 117sumpr 15091 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
119 prfi 8781 . . . . . 6 {𝐴, 𝐵} ∈ Fin
120119a1i 11 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → {𝐴, 𝐵} ∈ Fin)
12122adantl 482 . . . . . . . 8 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
12297adantr 481 . . . . . . . 8 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,)+∞))
123121, 122eqeltrd 2910 . . . . . . 7 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
124123ad4ant14 748 . . . . . 6 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
125 simp-4l 779 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
126 simpllr 772 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → ¬ 𝐸 = +∞)
12733adantll 710 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
128363adant2 1123 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
1291103adant3 1124 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,)+∞))
130128, 129eqeltrd 2910 . . . . . . 7 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,)+∞))
131125, 126, 127, 130syl3anc 1363 . . . . . 6 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
132124, 131pm2.61dan 809 . . . . 5 ((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,)+∞))
133120, 132sge0fsummpt 42549 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = Σ𝑘 ∈ {𝐴, 𝐵}𝐶)
13484, 98sseldi 3962 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℝ)
13584, 110sseldi 3962 . . . . . 6 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
136135adantlr 711 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
137 rexadd 12613 . . . . 5 ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℝ) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
138134, 136, 137syl2anc 584 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
139118, 133, 1383eqtr4d 2863 . . 3 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
14083, 139pm2.61dan 809 . 2 ((𝜑 ∧ ¬ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
14159, 140pm2.61dan 809 1 (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  {cpr 4559   class class class wbr 5057  cmpt 5137  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  Fincfn 8497  cc 10523  cr 10524  0cc0 10525   + caddc 10528  +∞cpnf 10660  -∞cmnf 10661  *cxr 10662   < clt 10663  cle 10664   +𝑒 cxad 12493  [,)cico 12728  [,]cicc 12729  Σcsu 15030  Σ^csumge0 42521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-xadd 12496  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-sumge0 42522
This theorem is referenced by:  sge0prle  42560  meadjun  42621  ovnsubadd2lem  42804
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