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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 41968 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | resubval 42054 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 −ℝ 𝐴) = (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)) | |
3 | 1, 2 | mpancom 686 | . 2 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) = (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)) |
4 | renegeu 42057 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | |
5 | riotacl 7393 | . . 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 2825 | 1 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃!wreu 3361 ℩crio 7374 (class class class)co 7419 ℝcr 11139 0cc0 11140 + caddc 11143 −ℝ cresub 42052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-addrcl 11201 ax-rnegex 11211 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-resub 42053 |
This theorem is referenced by: renegid 42060 reneg0addlid 42061 resubeulem1 42062 resubeulem2 42063 resubeu 42064 sn-00idlem2 42086 renegneg 42098 readdcan2 42099 renegid2 42100 sn-it0e0 42102 sn-negex12 42103 reixi 42109 rei4 42110 ipiiie0 42124 sn-0tie0 42126 zaddcomlem 42138 renegmulnnass 42140 zmulcomlem 42142 zmulcom 42143 mulgt0b2d 42147 sn-0lt1 42149 sn-inelr 42152 cnreeu 42155 |
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