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Theorem sge0pr 39905
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 12195 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
2 sge0pr.e . . . . . . 7 (𝜑𝐸 ∈ (0[,]+∞))
31, 2sseldi 3586 . . . . . 6 (𝜑𝐸 ∈ ℝ*)
4 mnfxr 10041 . . . . . . . 8 -∞ ∈ ℝ*
54a1i 11 . . . . . . 7 (𝜑 → -∞ ∈ ℝ*)
6 0xr 10031 . . . . . . . . 9 0 ∈ ℝ*
76a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℝ*)
8 mnflt0 11903 . . . . . . . . 9 -∞ < 0
98a1i 11 . . . . . . . 8 (𝜑 → -∞ < 0)
10 pnfxr 10037 . . . . . . . . . 10 +∞ ∈ ℝ*
1110a1i 11 . . . . . . . . 9 (𝜑 → +∞ ∈ ℝ*)
12 iccgelb 12169 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝐸 ∈ (0[,]+∞)) → 0 ≤ 𝐸)
137, 11, 2, 12syl3anc 1323 . . . . . . . 8 (𝜑 → 0 ≤ 𝐸)
145, 7, 3, 9, 13xrltletrd 11936 . . . . . . 7 (𝜑 → -∞ < 𝐸)
155, 3, 14xrgtned 38989 . . . . . 6 (𝜑𝐸 ≠ -∞)
16 xaddpnf2 12000 . . . . . 6 ((𝐸 ∈ ℝ*𝐸 ≠ -∞) → (+∞ +𝑒 𝐸) = +∞)
173, 15, 16syl2anc 692 . . . . 5 (𝜑 → (+∞ +𝑒 𝐸) = +∞)
1817eqcomd 2632 . . . 4 (𝜑 → +∞ = (+∞ +𝑒 𝐸))
1918adantr 481 . . 3 ((𝜑𝐷 = +∞) → +∞ = (+∞ +𝑒 𝐸))
20 prex 4875 . . . . 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 2704 . . . . . . . 8 ((𝜑𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
2726adantlr 750 . . . . . . 7 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
28 simpll 789 . . . . . . . 8 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
29 simpl 473 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ∈ {𝐴, 𝐵})
30 neqne 2804 . . . . . . . . . . 11 𝑘 = 𝐴𝑘𝐴)
3130adantl 482 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘𝐴)
32 elprn1 39256 . . . . . . . . . 10 ((𝑘 ∈ {𝐴, 𝐵} ∧ 𝑘𝐴) → 𝑘 = 𝐵)
3329, 31, 32syl2anc 692 . . . . . . . . 9 ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
3433adantll 749 . . . . . . . 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 2704 . . . . . . . 8 ((𝜑𝑘 = 𝐵) → 𝐶 ∈ (0[,]+∞))
3928, 34, 38syl2anc 692 . . . . . . 7 (((𝜑𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞))
4027, 39pm2.61dan 831 . . . . . 6 ((𝜑𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,]+∞))
41 eqid 2626 . . . . . 6 (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)
4240, 41fmptd 6341 . . . . 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 2632 . . . . . 6 (𝐷 = +∞ → +∞ = 𝐷)
4645adantl 482 . . . . 5 ((𝜑𝐷 = +∞) → +∞ = 𝐷)
47 prid1g 4270 . . . . . . . 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 38818 . . . . . . . 8 (𝜑 → ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = {𝐷, 𝐸})
5251eqcomd 2632 . . . . . . 7 (𝜑 → {𝐷, 𝐸} = ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5348, 52eleqtrd 2706 . . . . . 6 (𝜑𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5453adantr 481 . . . . 5 ((𝜑𝐷 = +∞) → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5546, 54eqeltrd 2704 . . . 4 ((𝜑𝐷 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
5621, 43, 55sge0pnfval 39884 . . 3 ((𝜑𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
57 oveq1 6612 . . . 4 (𝐷 = +∞ → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
5857adantl 482 . . 3 ((𝜑𝐷 = +∞) → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸))
5919, 56, 583eqtr4d 2670 . 2 ((𝜑𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
601, 24sseldi 3586 . . . . . . . 8 (𝜑𝐷 ∈ ℝ*)
61 iccgelb 12169 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝐷 ∈ (0[,]+∞)) → 0 ≤ 𝐷)
627, 11, 24, 61syl3anc 1323 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝐷)
635, 7, 60, 9, 62xrltletrd 11936 . . . . . . . . 9 (𝜑 → -∞ < 𝐷)
645, 60, 63xrgtned 38989 . . . . . . . 8 (𝜑𝐷 ≠ -∞)
65 xaddpnf1 11999 . . . . . . . 8 ((𝐷 ∈ ℝ*𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞)
6660, 64, 65syl2anc 692 . . . . . . 7 (𝜑 → (𝐷 +𝑒 +∞) = +∞)
6766eqcomd 2632 . . . . . 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 2632 . . . . . . . 8 (𝐸 = +∞ → +∞ = 𝐸)
7372adantl 482 . . . . . . 7 ((𝜑𝐸 = +∞) → +∞ = 𝐸)
74 prid2g 4271 . . . . . . . . . 10 (𝐸 ∈ (0[,]+∞) → 𝐸 ∈ {𝐷, 𝐸})
752, 74syl 17 . . . . . . . . 9 (𝜑𝐸 ∈ {𝐷, 𝐸})
7675, 52eleqtrd 2706 . . . . . . . 8 (𝜑𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7776adantr 481 . . . . . . 7 ((𝜑𝐸 = +∞) → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7873, 77eqeltrd 2704 . . . . . 6 ((𝜑𝐸 = +∞) → +∞ ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶))
7969, 70, 78sge0pnfval 39884 . . . . 5 ((𝜑𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞)
80 oveq2 6613 . . . . . 6 (𝐸 = +∞ → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
8180adantl 482 . . . . 5 ((𝜑𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒 +∞))
8268, 79, 813eqtr4d 2670 . . . 4 ((𝜑𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
8382adantlr 750 . . 3 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
84 rge0ssre 12219 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
85 ax-resscn 9938 . . . . . . . 8 ℝ ⊆ ℂ
8684, 85sstri 3597 . . . . . . 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 11908 . . . . . . . . . . . 12 (𝐷 ∈ ℝ*𝐷 ≤ +∞)
9260, 91syl 17 . . . . . . . . . . 11 (𝜑𝐷 ≤ +∞)
9392adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≤ +∞)
9444necon3bi 2822 . . . . . . . . . . 11 𝐷 = +∞ → 𝐷 ≠ +∞)
9594adantl 482 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≠ +∞)
9689, 88, 93, 95xrleneltd 38990 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 < +∞)
9787, 88, 89, 90, 96elicod 12163 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ (0[,)+∞))
9897adantr 481 . . . . . . 7 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ (0[,)+∞))
9986, 98sseldi 3586 . . . . . 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 11908 . . . . . . . . . . . 12 (𝐸 ∈ ℝ*𝐸 ≤ +∞)
1053, 104syl 17 . . . . . . . . . . 11 (𝜑𝐸 ≤ +∞)
106105adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≤ +∞)
10771necon3bi 2822 . . . . . . . . . . 11 𝐸 = +∞ → 𝐸 ≠ +∞)
108107adantl 482 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≠ +∞)
109102, 101, 106, 108xrleneltd 38990 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 < +∞)
110100, 101, 102, 103, 109elicod 12163 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ (0[,)+∞))
11186, 110sseldi 3586 . . . . . . 7 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
112111adantlr 750 . . . . . 6 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ)
11399, 112jca 554 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
11449, 50jca 554 . . . . . 6 (𝜑 → (𝐴𝑉𝐵𝑊))
115114ad2antrr 761 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐴𝑉𝐵𝑊))
116 sge0pr.ab . . . . . 6 (𝜑𝐴𝐵)
117116ad2antrr 761 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐴𝐵)
11822, 35, 113, 115, 117sumpr 14402 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
119 prfi 8180 . . . . . 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 2704 . . . . . . 7 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
124123ad4ant14 1290 . . . . . 6 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
125 simp-4l 805 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑)
126 simpllr 798 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → ¬ 𝐸 = +∞)
12733adantll 749 . . . . . . 7 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵)
128363adant2 1078 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
1291103adant3 1079 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,)+∞))
130128, 129eqeltrd 2704 . . . . . . 7 ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,)+∞))
131125, 126, 127, 130syl3anc 1323 . . . . . 6 (((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞))
132124, 131pm2.61dan 831 . . . . 5 ((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,)+∞))
133120, 132sge0fsummpt 39901 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = Σ𝑘 ∈ {𝐴, 𝐵}𝐶)
13484, 98sseldi 3586 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℝ)
13584, 110sseldi 3586 . . . . . 6 ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
136135adantlr 750 . . . . 5 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ)
137 rexadd 12005 . . . . 5 ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℝ) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
138134, 136, 137syl2anc 692 . . . 4 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸))
139118, 133, 1383eqtr4d 2670 . . 3 (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
14083, 139pm2.61dan 831 . 2 ((𝜑 ∧ ¬ 𝐷 = +∞) → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
14159, 140pm2.61dan 831 1 (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  Vcvv 3191  {cpr 4155   class class class wbr 4618  cmpt 4678  ran crn 5080  wf 5846  cfv 5850  (class class class)co 6605  Fincfn 7900  cc 9879  cr 9880  0cc0 9881   + caddc 9884  +∞cpnf 10016  -∞cmnf 10017  *cxr 10018   < clt 10019  cle 10020   +𝑒 cxad 11888  [,)cico 12116  [,]cicc 12117  Σcsu 14345  Σ^csumge0 39873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-xadd 11891  df-ico 12120  df-icc 12121  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-sum 14346  df-sumge0 39874
This theorem is referenced by:  sge0prle  39912  meadjun  39973  ovnsubadd2lem  40153
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