elbigof - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > elbigof		Structured version Visualization version GIF version

Theorem elbigof 44634

Description: A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)

**Assertion**
Ref	Expression
elbigof	⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)

Proof of Theorem elbigof Dummy variables 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elbigo 44631	. 2 ⊢ (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑_pm ℝ) ∧ 𝐺 ∈ (ℝ ↑_pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
2		reex 10628	. . . . 5 ⊢ ℝ ∈ V
3	2, 2	elpm2 8438	. . . 4 ⊢ (𝐹 ∈ (ℝ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
4	3	simplbi 500	. . 3 ⊢ (𝐹 ∈ (ℝ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℝ)
5	4	3ad2ant1 1129	. 2 ⊢ ((𝐹 ∈ (ℝ ↑_pm ℝ) ∧ 𝐺 ∈ (ℝ ↑_pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))) → 𝐹:dom 𝐹⟶ℝ)
6	1, 5	sylbi 219	1 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_pm cpm 8407 ℝcr 10536 · cmul 10542 +∞cpnf 10672 ≤ cle 10676 [,)cico 12741 Οcbigo 44627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-pm 8409 df-bigo 44628

This theorem is referenced by: (None)

W3C validator