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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrsinvgval | Structured version Visualization version GIF version |
Description: The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13252 and df-xrs 17475), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
Ref | Expression |
---|---|
xrsinvgval | ⊢ (𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrsbas 21298 | . . 3 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
2 | xrsadd 21299 | . . 3 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
3 | xrs0 32715 | . . 3 ⊢ 0 = (0g‘ℝ*𝑠) | |
4 | eqid 2727 | . . 3 ⊢ (invg‘ℝ*𝑠) = (invg‘ℝ*𝑠) | |
5 | 1, 2, 3, 4 | grpinvval 18928 | . 2 ⊢ (𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = (℩𝑥 ∈ ℝ* (𝑥 +𝑒 𝐵) = 0)) |
6 | xnegcl 13216 | . . 3 ⊢ (𝐵 ∈ ℝ* → -𝑒𝐵 ∈ ℝ*) | |
7 | xaddeq0 32507 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑥 +𝑒 𝐵) = 0 ↔ 𝑥 = -𝑒𝐵)) | |
8 | 7 | ancoms 458 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → ((𝑥 +𝑒 𝐵) = 0 ↔ 𝑥 = -𝑒𝐵)) |
9 | 6, 8 | riota5 7400 | . 2 ⊢ (𝐵 ∈ ℝ* → (℩𝑥 ∈ ℝ* (𝑥 +𝑒 𝐵) = 0) = -𝑒𝐵) |
10 | 5, 9 | eqtrd 2767 | 1 ⊢ (𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 ℩crio 7369 (class class class)co 7414 0cc0 11130 ℝ*cxr 11269 -𝑒cxne 13113 +𝑒 cxad 13114 ℝ*𝑠cxrs 17473 invgcminusg 18882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-xneg 13116 df-xadd 13117 df-fz 13509 df-struct 17107 df-slot 17142 df-ndx 17154 df-base 17172 df-plusg 17237 df-mulr 17238 df-tset 17243 df-ple 17244 df-ds 17246 df-0g 17414 df-xrs 17475 df-minusg 18885 |
This theorem is referenced by: xrsmulgzz 32718 |
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