xnegnegi - Mathbox for Glauco Siliprandi

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Theorem xnegnegi 45401

Description: Extended real version of negneg 11563. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypothesis**
Ref	Expression
xnegnegi.1	⊢ 𝐴 ∈ ℝ^*

**Assertion**
Ref	Expression
xnegnegi	⊢ -_𝑒-_𝑒𝐴 = 𝐴

**Proof of Theorem xnegnegi**
Step	Hyp	Ref	Expression
1		xnegnegi.1	. 2 ⊢ 𝐴 ∈ ℝ^*
2		xnegneg 13259	. 2 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ -_𝑒-_𝑒𝐴 = 𝐴

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 ∈ wcel 2107 ℝ^*cxr 11298 -_𝑒cxne 13155

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5303 ax-nul 5313 ax-pow 5372 ax-pr 5439 ax-un 7758 ax-cnex 11215 ax-resscn 11216 ax-1cn 11217 ax-icn 11218 ax-addcl 11219 ax-addrcl 11220 ax-mulcl 11221 ax-mulrcl 11222 ax-mulcom 11223 ax-addass 11224 ax-mulass 11225 ax-distr 11226 ax-i2m1 11227 ax-1ne0 11228 ax-1rid 11229 ax-rnegex 11230 ax-rrecex 11231 ax-cnre 11232 ax-pre-lttri 11233 ax-pre-lttrn 11234 ax-pre-ltadd 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1541 df-fal 1551 df-ex 1778 df-nf 1782 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3435 df-v 3481 df-sbc 3793 df-csb 3910 df-dif 3967 df-un 3969 df-in 3971 df-ss 3981 df-nul 4341 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4914 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5584 df-po 5598 df-so 5599 df-xp 5696 df-rel 5697 df-cnv 5698 df-co 5699 df-dm 5700 df-rn 5701 df-res 5702 df-ima 5703 df-iota 6519 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7392 df-ov 7438 df-oprab 7439 df-mpo 7440 df-er 8750 df-en 8991 df-dom 8992 df-sdom 8993 df-pnf 11301 df-mnf 11302 df-xr 11303 df-ltxr 11304 df-sub 11498 df-neg 11499 df-xneg 13158

This theorem is referenced by: smfliminflem 46797