![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xrsadd | Structured version Visualization version GIF version |
Description: The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsadd | ⊢ +𝑒 = (+g‘ℝ*𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xaddf 13201 | . . 3 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | |
2 | xrex 12969 | . . . 4 ⊢ ℝ* ∈ V | |
3 | 2, 2 | xpex 7734 | . . 3 ⊢ (ℝ* × ℝ*) ∈ V |
4 | fex2 7918 | . . 3 ⊢ (( +𝑒 :(ℝ* × ℝ*)⟶ℝ* ∧ (ℝ* × ℝ*) ∈ V ∧ ℝ* ∈ V) → +𝑒 ∈ V) | |
5 | 1, 3, 2, 4 | mp3an 1457 | . 2 ⊢ +𝑒 ∈ V |
6 | df-xrs 17449 | . . 3 ⊢ ℝ*𝑠 = ({〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∪ {〈(TopSet‘ndx), (ordTop‘ ≤ )〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))〉}) | |
7 | 6 | odrngplusg 17351 | . 2 ⊢ ( +𝑒 ∈ V → +𝑒 = (+g‘ℝ*𝑠)) |
8 | 5, 7 | ax-mp 5 | 1 ⊢ +𝑒 = (+g‘ℝ*𝑠) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3466 ifcif 4521 class class class wbr 5139 × cxp 5665 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 ℝ*cxr 11245 ≤ cle 11247 -𝑒cxne 13087 +𝑒 cxad 13088 ·e cxmu 13089 +gcplusg 17198 ordTopcordt 17446 ℝ*𝑠cxrs 17447 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-xadd 13091 df-fz 13483 df-struct 17081 df-slot 17116 df-ndx 17128 df-base 17146 df-plusg 17211 df-mulr 17212 df-tset 17217 df-ple 17218 df-ds 17220 df-xrs 17449 |
This theorem is referenced by: xrsmgm 21266 xrsnsgrp 21267 xrs1mnd 21269 xrs10 21270 xrs1cmn 21271 xrge0subm 21272 imasdsf1olem 24203 xrge0gsumle 24673 xrs0 32646 xrsinvgval 32648 xrsmulgzz 32649 xrge0plusg 32656 xrge0tmdALT 33418 esumpfinvallem 33564 sge0tsms 45606 |
Copyright terms: Public domain | W3C validator |