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Mirrors > Home > MPE Home > Th. List > xaddid2 | Structured version Visualization version GIF version |
Description: Extended real version of addid2 10431. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddid2 | ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10298 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xaddcom 12284 | . . 3 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (0 +𝑒 𝐴) = (𝐴 +𝑒 0)) | |
3 | 1, 2 | mpan 708 | . 2 ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = (𝐴 +𝑒 0)) |
4 | xaddid1 12285 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | |
5 | 3, 4 | eqtrd 2794 | 1 ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 (class class class)co 6814 0cc0 10148 ℝ*cxr 10285 +𝑒 cxad 12157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-xadd 12160 |
This theorem is referenced by: xaddge0 12301 xsubge0 12304 xadddi2 12340 xrs1mnd 20006 xrs10 20007 imasdsf1olem 22399 stdbdxmet 22541 xaddeq0 29848 xrs0 30005 xrsmulgzz 30008 xrge0adddir 30022 xrge0npcan 30024 metideq 30266 esumrnmpt2 30460 esumpfinvallem 30466 0elcarsg 30699 carsgclctunlem3 30712 xaddid2d 40051 sge0tsms 41118 meadjun 41200 caragencmpl 41273 |
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