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Mirrors > Home > MPE Home > Th. List > xaddlid | Structured version Visualization version GIF version |
Description: Extended real version of addlid 11404. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddlid | ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 11268 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xaddcom 13226 | . . 3 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (0 +𝑒 𝐴) = (𝐴 +𝑒 0)) | |
3 | 1, 2 | mpan 687 | . 2 ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = (𝐴 +𝑒 0)) |
4 | xaddrid 13227 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | |
5 | 3, 4 | eqtrd 2771 | 1 ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 (class class class)co 7412 0cc0 11116 ℝ*cxr 11254 +𝑒 cxad 13097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-xadd 13100 |
This theorem is referenced by: xaddge0 13244 xsubge0 13247 xadddi2 13283 xrs1mnd 21273 xrs10 21274 imasdsf1olem 24200 stdbdxmet 24345 xaddeq0 32401 xrs0 32611 xrsmulgzz 32614 xrge0adddir 32628 xrge0npcan 32630 metideq 33339 esumrnmpt2 33532 esumpfinvallem 33538 0elcarsg 33772 carsgclctunlem3 33785 xaddlidd 44491 sge0tsms 45558 meadjun 45640 caragencmpl 45713 |
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