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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpval | Structured version Visualization version GIF version |
Description: Define the value of the decimal point operator. See df-dp 29596. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
dpval | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dp2 29578 | . . 3 ⊢ _𝑥𝑦 = (𝑥 + (𝑦 / ;10)) | |
2 | oveq1 6657 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (𝑦 / ;10)) = (𝐴 + (𝑦 / ;10))) | |
3 | 1, 2 | syl5eq 2668 | . 2 ⊢ (𝑥 = 𝐴 → _𝑥𝑦 = (𝐴 + (𝑦 / ;10))) |
4 | oveq1 6657 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 / ;10) = (𝐵 / ;10)) | |
5 | 4 | oveq2d 6666 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = (𝐴 + (𝐵 / ;10))) |
6 | df-dp2 29578 | . . 3 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
7 | 5, 6 | syl6eqr 2674 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = _𝐴𝐵) |
8 | df-dp 29596 | . 2 ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦) | |
9 | 6 | ovexi 6679 | . 2 ⊢ _𝐴𝐵 ∈ V |
10 | 3, 7, 8, 9 | ovmpt2 6796 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 / cdiv 10684 ℕ0cn0 11292 ;cdc 11493 _cdp2 29577 .cdp 29595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-dp2 29578 df-dp 29596 |
This theorem is referenced by: dpcl 29598 dpfrac1 29599 dpfrac1OLD 29600 dpval2 29601 dpmul1000 29607 dpadd2 29618 |
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