dpval3rp - Mathbox for Thierry Arnoux

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Theorem dpval3rp 32623

Description: Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)

**Hypotheses**
Ref	Expression
dpval3rp.a	⊢ 𝐴 ∈ ℕ₀
dpval3rp.b	⊢ 𝐵 ∈ ℝ⁺

**Assertion**
Ref	Expression
dpval3rp	⊢ (𝐴.𝐵) = _𝐴𝐵

**Proof of Theorem dpval3rp**
Step	Hyp	Ref	Expression
1		dpval3rp.a	. 2 ⊢ 𝐴 ∈ ℕ₀
2		dpval3rp.b	. . 3 ⊢ 𝐵 ∈ ℝ⁺
3		rpre 13014	. . 3 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
4	2, 3	ax-mp 5	. 2 ⊢ 𝐵 ∈ ℝ
5	1, 4	dpval3 32617	1 ⊢ (𝐴.𝐵) = _𝐴𝐵

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7420 ℝcr 11137 ℕ₀cn0 12502 ℝ⁺crp 13006 cdp2 32594 .cdp 32611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-rp 13007 df-dp2 32595 df-dp 32612

This theorem is referenced by: dp0h 32625 rpdpcl 32626 dplt 32627 dpltc 32630 dpexpp1 32631

	
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