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Theorem sge0pr 39287
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 12127 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
2 sge0pr.e . . . . . . 7 (𝜑𝐸 ∈ (0[,]+∞))
31, 2sseldi 3566 . . . . . 6 (𝜑𝐸 ∈ ℝ*)
4 mnfxr 9975 . . . . . . . 8 -∞ ∈ ℝ*
54a1i 11 . . . . . . 7 (𝜑 → -∞ ∈ ℝ*)
6 0xr 9965 . . . . . . . . 9 0 ∈ ℝ*
76a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℝ*)
8 mnflt0 11835 . . . . . . . . 9 -∞ < 0
98a1i 11 . . . . . . . 8 (𝜑 → -∞ < 0)
10 pnfxr 9971 . . . . . . . . . 10 +∞ ∈ ℝ*
1110a1i 11 . . . . . . . . 9 (𝜑 → +∞ ∈ ℝ*)
12 iccgelb 12101 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝐸 ∈ (0[,]+∞)) → 0 ≤ 𝐸)
137, 11, 2, 12syl3anc 1318 . . . . . . . 8 (𝜑 → 0 ≤ 𝐸)
145, 7, 3, 9, 13xrltletrd 11868 . . . . . . 7 (𝜑 → -∞ < 𝐸)
155, 3, 14xrgtned 38479 . . . . . 6 (𝜑𝐸 ≠ -∞)
16 xaddpnf2 11932 . . . . . 6 ((𝐸 ∈ ℝ*𝐸 ≠ -∞) → (+∞ +𝑒 𝐸) = +∞)
173, 15, 16syl2anc 691 . . . . 5 (𝜑 → (+∞ +𝑒 𝐸) = +∞)
1817eqcomd 2616 . . . 4 (𝜑 → +∞ = (+∞ +𝑒 𝐸))
1918adantr 480 . . 3 ((𝜑𝐷 = +∞) → +∞ = (+∞ +𝑒 𝐸))
20 prex 4836 . . . . 5 {𝐴, 𝐵} ∈ V
2120a1i 11 . . . 4 ((𝜑𝐷 = +∞) → {𝐴, 𝐵} ∈ V)
22 sge0pr.cd . . . . . . . . . 10 (𝑘 = 𝐴𝐶 = 𝐷)
2322adantl 481 . . . . . . . . 9 ((𝜑𝑘 = 𝐴) → 𝐶 = 𝐷)
24 sge0pr.d . . . . . . . . . 10 (𝜑𝐷 ∈ (0[,]+∞))
2524adantr 480 . . . . . . . . 9 ((𝜑𝑘 = 𝐴) → 𝐷 ∈ (0[,]+∞))
2623, 25eqeltrd 2688 . . . . . . . 8 ((𝜑𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
2726adantlr 747 . . . . . . 7 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
28 simpll 786 . . . . . . . 8 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
29 simpl 472 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ∈ {𝐴, 𝐵})
30 neqne 2790 . . . . . . . . . . 11 𝑘 = 𝐴𝑘𝐴)
3130adantl 481 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘𝐴)
32 elprn1 38700 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ 𝑘𝐴) → 𝑘 = 𝐵)
3329, 31, 32syl2anc 691 . . . . . . . . 9 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
3433adantll 746 . . . . . . . 8 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
35 sge0pr.ce . . . . . . . . . 10 (𝑘 = 𝐵𝐶 = 𝐸)
3635adantl 481 . . . . . . . . 9 ((𝜑𝑘 = 𝐵) → 𝐶 = 𝐸)
372adantr 480 . . . . . . . . 9 ((𝜑𝑘 = 𝐵) → 𝐸 ∈ (0[,]+∞))
3836, 37eqeltrd 2688 . . . . . . . 8 ((𝜑𝑘 = 𝐵) → 𝐶 ∈ (0[,]+∞))
3928, 34, 38syl2anc 691 . . . . . . 7 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
4027, 39pm2.61dan 828 . . . . . 6 ((𝜑𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,]+∞))
41 eqid 2610 . . . . . 6 (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)
4240, 41fmptd 6292 . . . . 5 (𝜑 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
4342adantr 480 . . . 4 ((𝜑𝐷 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
44 id 22 . . . . . . 7 (𝐷 = +∞ → 𝐷 = +∞)
4544eqcomd 2616 . . . . . 6 (𝐷 = +∞ → +∞ = 𝐷)
4645adantl 481 . . . . 5 ((𝜑𝐷 = +∞) → +∞ = 𝐷)
47 prid1g 4239 . . . . . . . 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 38353 . . . . . . . 8 (𝜑 → ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = {𝐷, 𝐸})
5251eqcomd 2616 . . . . . . 7 (𝜑 → {𝐷, 𝐸} = ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5348, 52eleqtrd 2690 . . . . . 6 (𝜑𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5453adantr 480 . . . . 5 ((𝜑𝐷 = +∞) → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5546, 54eqeltrd 2688 . . . 4 ((𝜑𝐷 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5621, 43, 55sge0pnfval 39266 . . 3 ((𝜑𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
57 oveq1 6556 . . . 4 (𝐷 = +∞ → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
5857adantl 481 . . 3 ((𝜑𝐷 = +∞) → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
5919, 56, 583eqtr4d 2654 . 2 ((𝜑𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
601, 24sseldi 3566 . . . . . . . 8 (𝜑𝐷 ∈ ℝ*)
61 iccgelb 12101 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷)
627, 11, 24, 61syl3anc 1318 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝐷)
635, 7, 60, 9, 62xrltletrd 11868 . . . . . . . . 9 (𝜑 → -∞ < 𝐷)
645, 60, 63xrgtned 38479 . . . . . . . 8 (𝜑𝐷 ≠ -∞)
65 xaddpnf1 11931 . . . . . . . 8 ((𝐷 ∈ ℝ*𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞)
6660, 64, 65syl2anc 691 . . . . . . 7 (𝜑 → (𝐷 +𝑒 +∞) = +∞)
6766eqcomd 2616 . . . . . 6 (𝜑 → +∞ = (𝐷 +𝑒 +∞))
6867adantr 480 . . . . 5 ((𝜑𝐸 = +∞) → +∞ = (𝐷 +𝑒 +∞))
6920a1i 11 . . . . . 6 ((𝜑𝐸 = +∞) → {𝐴, 𝐵} ∈ V)
7042adantr 480 . . . . . 6 ((𝜑𝐸 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞))
71 id 22 . . . . . . . . 9 (𝐸 = +∞ → 𝐸 = +∞)
7271eqcomd 2616 . . . . . . . 8 (𝐸 = +∞ → +∞ = 𝐸)
7372adantl 481 . . . . . . 7 ((𝜑𝐸 = +∞) → +∞ = 𝐸)
74 prid2g 4240 . . . . . . . . . 10 (𝐸 ∈ (0[,]+∞) → 𝐸 ∈ {𝐷, 𝐸})
752, 74syl 17 . . . . . . . . 9 (𝜑𝐸 ∈ {𝐷, 𝐸})
7675, 52eleqtrd 2690 . . . . . . . 8 (𝜑𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7776adantr 480 . . . . . . 7 ((𝜑𝐸 = +∞) → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7873, 77eqeltrd 2688 . . . . . 6 ((𝜑𝐸 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7969, 70, 78sge0pnfval 39266 . . . . 5 ((𝜑𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
80 oveq2 6557 . . . . . 6 (𝐸 = +∞ → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
8180adantl 481 . . . . 5 ((𝜑𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
8268, 79, 813eqtr4d 2654 . . . 4 ((𝜑𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
8382adantlr 747 . . 3 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
84 rge0ssre 12151 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
85 ax-resscn 9872 . . . . . . . 8 ℝ ⊆ ℂ
8684, 85sstri 3577 . . . . . . 7 (0[,)+∞) ⊆ ℂ
876a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ∈ ℝ*)
8810a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → +∞ ∈ ℝ*)
8960adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ ℝ*)
9062adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ≤ 𝐷)
91 pnfge 11840 . . . . . . . . . . . 12 (𝐷 ∈ ℝ*𝐷 ≤ +∞)
9260, 91syl 17 . . . . . . . . . . 11 (𝜑𝐷 ≤ +∞)
9392adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≤ +∞)
9444necon3bi 2808 . . . . . . . . . . 11 𝐷 = +∞ → 𝐷 ≠ +∞)
9594adantl 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≠ +∞)
9689, 88, 93, 95xrleneltd 38480 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 < +∞)
9787, 88, 89, 90, 96elicod 12095 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ (0[,)+∞))
9897adantr 480 . . . . . . 7 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ (0[,)+∞))
9986, 98sseldi 3566 . . . . . 6 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℂ)
1006a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ∈ ℝ*)
10110a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → +∞ ∈ ℝ*)
1023adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ*)
10313adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ≤ 𝐸)
104 pnfge 11840 . . . . . . . . . . . 12 (𝐸 ∈ ℝ*𝐸 ≤ +∞)
1053, 104syl 17 . . . . . . . . . . 11 (𝜑𝐸 ≤ +∞)
106105adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≤ +∞)
10771necon3bi 2808 . . . . . . . . . . 11 𝐸 = +∞ → 𝐸 ≠ +∞)
108107adantl 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≠ +∞)
109102, 101, 106, 108xrleneltd 38480 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 < +∞)
110100, 101, 102, 103, 109elicod 12095 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ (0[,)+∞))
11186, 110sseldi 3566 . . . . . . 7 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
112111adantlr 747 . . . . . 6 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
11399, 112jca 553 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
11449, 50jca 553 . . . . . 6 (𝜑 → (𝐴𝑉𝐵𝑊))
115114ad2antrr 758 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐴𝑉𝐵𝑊))
116 sge0pr.ab . . . . . 6 (𝜑𝐴𝐵)
117116ad2antrr 758 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐴𝐵)
11822, 35, 113, 115, 117sumpr 14321 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
119 prfi 8120 . . . . . 6 {𝐴, 𝐵} ∈ Fin
120119a1i 11 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → {𝐴, 𝐵} ∈ Fin)
12122adantl 481 . . . . . . . 8 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
12297adantr 480 . . . . . . . 8 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,)+∞))
123121, 122eqeltrd 2688 . . . . . . 7 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
124123ad4ant14 1285 . . . . . 6 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
125 simp-4l 802 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
126 simpllr 795 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → ¬ 𝐸 = +∞)
12733adantll 746 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
128363adant2 1073 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
1291103adant3 1074 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,)+∞))
130128, 129eqeltrd 2688 . . . . . . 7 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,)+∞))
131125, 126, 127, 130syl3anc 1318 . . . . . 6 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
132124, 131pm2.61dan 828 . . . . 5 ((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,)+∞))
133120, 132sge0fsummpt 39283 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = Σ𝑘 ∈ {𝐴, 𝐵}𝐶)
13484, 98sseldi 3566 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℝ)
13584, 110sseldi 3566 . . . . . 6 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
136135adantlr 747 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
137 rexadd 11937 . . . . 5 ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℝ) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
138134, 136, 137syl2anc 691 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
139118, 133, 1383eqtr4d 2654 . . 3 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
14083, 139pm2.61dan 828 . 2 ((𝜑 ∧ ¬ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
14159, 140pm2.61dan 828 1 (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  {cpr 4127   class class class wbr 4583  cmpt 4643  ran crn 5039  wf 5800  cfv 5804  (class class class)co 6549  Fincfn 7841  cc 9813  cr 9814  0cc0 9815   + caddc 9818  +∞cpnf 9950  -∞cmnf 9951  *cxr 9952   < clt 9953  cle 9954   +𝑒 cxad 11820  [,)cico 12048  [,]cicc 12049  Σcsu 14264  Σ^csumge0 39255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-sumge0 39256
This theorem is referenced by:  sge0prle  39294  meadjun  39355  ovnsubadd2lem  39535
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