xlenegcon1 - Mathbox for Glauco Siliprandi

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Theorem xlenegcon1 44949

Description: Extended real version of lenegcon1 11748. (Contributed by Glauco Siliprandi, 23-Apr-2023.)

**Assertion**
Ref	Expression
xlenegcon1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ≤ 𝐵 ↔ -_𝑒𝐵 ≤ 𝐴))

**Proof of Theorem xlenegcon1**
Step	Hyp	Ref	Expression
1		xnegcl 13224	. . 3 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
2		xleneg 13229	. . 3 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ≤ 𝐵 ↔ -_𝑒𝐵 ≤ -_𝑒-_𝑒𝐴))
3	1, 2	sylan 578	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ≤ 𝐵 ↔ -_𝑒𝐵 ≤ -_𝑒-_𝑒𝐴))
4		xnegneg 13225	. . . 4 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
5	4	breq2d 5160	. . 3 ⊢ (𝐴 ∈ ℝ^* → (-_𝑒𝐵 ≤ -_𝑒-_𝑒𝐴 ↔ -_𝑒𝐵 ≤ 𝐴))
6	5	adantr 479	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐵 ≤ -_𝑒-_𝑒𝐴 ↔ -_𝑒𝐵 ≤ 𝐴))
7	3, 6	bitrd 278	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ≤ 𝐵 ↔ -_𝑒𝐵 ≤ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5148 ℝ^*cxr 11277 ≤ cle 11279 -_𝑒cxne 13121

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 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214

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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-xneg 13124

This theorem is referenced by: xlimpnfxnegmnf 45282

	
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