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Mirrors > Home > MPE Home > Th. List > Mathboxes > rernegcl | Structured version Visualization version GIF version |
Description: Closure law for negative reals. (Contributed by Steven Nguyen, 7-Jan-2023.) |
Ref | Expression |
---|---|
rernegcl | ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elre0re 39462 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | resubval 39505 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 −ℝ 𝐴) = (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)) | |
3 | 1, 2 | mpancom 687 | . 2 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) = (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)) |
4 | renegeu 39508 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | |
5 | riotacl 7110 | . . 3 ⊢ (∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) ∈ ℝ) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) ∈ ℝ) |
7 | 3, 6 | eqeltrd 2890 | 1 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∃!wreu 3108 ℩crio 7092 (class class class)co 7135 ℝcr 10525 0cc0 10526 + caddc 10529 −ℝ cresub 39503 |
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 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-addrcl 10587 ax-rnegex 10597 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-resub 39504 |
This theorem is referenced by: renegid 39511 reneg0addid2 39512 resubeulem1 39513 resubeulem2 39514 resubeu 39515 sn-00idlem2 39537 renegneg 39549 readdcan2 39550 renegid2 39551 sn-it0e0 39552 sn-negex12 39553 reixi 39559 rei4 39560 ipiiie0 39574 sn-0tie0 39576 mulgt0b2d 39585 sn-0lt1 39587 sn-inelr 39590 cnreeu 39593 |
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