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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrsp0 | Structured version Visualization version GIF version |
Description: The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Proof shortened by AV, 28-Sep-2020.) |
Ref | Expression |
---|---|
xrsp0 | ⊢ -∞ = (0.‘ℝ*𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrsex 21265 | . . 3 ⊢ ℝ*𝑠 ∈ V | |
2 | xrsbas 21266 | . . . 4 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
3 | eqid 2724 | . . . 4 ⊢ (glb‘ℝ*𝑠) = (glb‘ℝ*𝑠) | |
4 | eqid 2724 | . . . 4 ⊢ (0.‘ℝ*𝑠) = (0.‘ℝ*𝑠) | |
5 | 2, 3, 4 | p0val 18388 | . . 3 ⊢ (ℝ*𝑠 ∈ V → (0.‘ℝ*𝑠) = ((glb‘ℝ*𝑠)‘ℝ*)) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (0.‘ℝ*𝑠) = ((glb‘ℝ*𝑠)‘ℝ*) |
7 | ssid 3997 | . . 3 ⊢ ℝ* ⊆ ℝ* | |
8 | xrslt 32669 | . . . 4 ⊢ < = (lt‘ℝ*𝑠) | |
9 | xrstos 32672 | . . . . 5 ⊢ ℝ*𝑠 ∈ Toset | |
10 | 9 | a1i 11 | . . . 4 ⊢ (ℝ* ⊆ ℝ* → ℝ*𝑠 ∈ Toset) |
11 | id 22 | . . . 4 ⊢ (ℝ* ⊆ ℝ* → ℝ* ⊆ ℝ*) | |
12 | 2, 8, 10, 11 | tosglb 32637 | . . 3 ⊢ (ℝ* ⊆ ℝ* → ((glb‘ℝ*𝑠)‘ℝ*) = inf(ℝ*, ℝ*, < )) |
13 | 7, 12 | ax-mp 5 | . 2 ⊢ ((glb‘ℝ*𝑠)‘ℝ*) = inf(ℝ*, ℝ*, < ) |
14 | xrinfm 32461 | . 2 ⊢ inf(ℝ*, ℝ*, < ) = -∞ | |
15 | 6, 13, 14 | 3eqtrri 2757 | 1 ⊢ -∞ = (0.‘ℝ*𝑠) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⊆ wss 3941 ‘cfv 6534 infcinf 9433 -∞cmnf 11245 ℝ*cxr 11246 < clt 11247 ℝ*𝑠cxrs 17451 glbcglb 18271 Tosetctos 18377 0.cp0 18384 |
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-rep 5276 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 ax-pre-sup 11185 |
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-rmo 3368 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-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-tset 17221 df-ple 17222 df-ds 17224 df-xrs 17453 df-proset 18256 df-poset 18274 df-plt 18291 df-glb 18308 df-toset 18378 df-p0 18386 |
This theorem is referenced by: (None) |
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