o1add2 - Metamath Proof Explorer

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Theorem o1add2 15033

Description: The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)

**Hypotheses**
Ref	Expression
o1add2.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
o1add2.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
o1add2.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))
o1add2.4	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))

**Assertion**
Ref	Expression
o1add2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥) 𝑉(𝑥)

**Proof of Theorem o1add2**
Step	Hyp	Ref	Expression
1		o1add2.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
2	1	ralrimiva 3113	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
3		dmmptg 6075	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
5		o1add2.3	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))
6		o1dm 14940	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
8	4, 7	eqsstrrd 3933	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9		reex 10671	. . . . 5 ⊢ ℝ ∈ V
10	9	ssex 5194	. . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
11	8, 10	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
12		o1add2.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
13		eqidd 2759	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
14		eqidd 2759	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶))
15	11, 1, 12, 13, 14	offval2 7429	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘_f + (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)))
16		o1add2.4	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
17		o1add 15023	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1)) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘_f + (𝑥 ∈ 𝐴 ↦ 𝐶)) ∈ 𝑂(1))
18	5, 16, 17	syl2anc 587	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘_f + (𝑥 ∈ 𝐴 ↦ 𝐶)) ∈ 𝑂(1))
19	15, 18	eqeltrrd 2853	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3070 Vcvv 3409 ⊆ wss 3860 ↦ cmpt 5115 dom cdm 5527 (class class class)co 7155 ∘_f cof 7408 ℝcr 10579 + caddc 10583 𝑂(1)co1 14896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 ax-pre-sup 10658

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-om 7585 df-2nd 7699 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-er 8304 df-pm 8424 df-en 8533 df-dom 8534 df-sdom 8535 df-sup 8944 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-div 11341 df-nn 11680 df-2 11742 df-3 11743 df-n0 11940 df-z 12026 df-uz 12288 df-rp 12436 df-ico 12790 df-seq 13424 df-exp 13485 df-cj 14511 df-re 14512 df-im 14513 df-sqrt 14647 df-abs 14648 df-o1 14900

This theorem is referenced by: dchrvmasumlem2 26186 dchrisum0lem1 26204 dchrisum0lem3 26207 mulog2sumlem2 26223 selberglem1 26233 selberg3 26247 selberg4 26249 selberg4r 26258 selberg34r 26259 pntrlog2bndlem2 26266

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