| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rr2sscn2 | Structured version Visualization version GIF version | ||
| Description: The cartesian square of ℝ is a subset of the cartesian square of ℂ. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
| Ref | Expression |
|---|---|
| rr2sscn2 | ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-resscn 11212 | . 2 ⊢ ℝ ⊆ ℂ | |
| 2 | xpss12 5700 | . 2 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ)) | |
| 3 | 1, 1, 2 | mp2an 692 | 1 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ wss 3951 × cxp 5683 ℂcc 11153 ℝcr 11154 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: ovolval2lem 46658 ovolval2 46659 ovolval3 46662 |
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