xneg11 - Metamath Proof Explorer

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Theorem xneg11 13229

Description: Extended real version of neg11 11543. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xneg11	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒𝐵 ↔ 𝐴 = 𝐵))

**Proof of Theorem xneg11**
Step	Hyp	Ref	Expression
1		xnegeq 13221	. . 3 ⊢ (-_𝑒𝐴 = -_𝑒𝐵 → -_𝑒-_𝑒𝐴 = -_𝑒-_𝑒𝐵)
2		xnegneg 13228	. . . 4 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
3		xnegneg 13228	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
4	2, 3	eqeqan12d 2739	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒-_𝑒𝐴 = -_𝑒-_𝑒𝐵 ↔ 𝐴 = 𝐵))
5	1, 4	imbitrid 243	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒𝐵 → 𝐴 = 𝐵))
6		xnegeq 13221	. 2 ⊢ (𝐴 = 𝐵 → -_𝑒𝐴 = -_𝑒𝐵)
7	5, 6	impbid1 224	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒𝐵 ↔ 𝐴 = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ℝ^*cxr 11279 -_𝑒cxne 13124

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-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 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-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-xr 11284 df-ltxr 11285 df-sub 11478 df-neg 11479 df-xneg 13127

This theorem is referenced by: xaddass2 13264 xmulneg1 13283 xmulpnf1n 13292 xmulasslem 13299 xadddi 13309 xadddi2 13311 xrsxmet 24769 liminfvalxr 45309 liminflbuz2 45341 liminfpnfuz 45342

	
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