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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrsp1 | Structured version Visualization version GIF version |
Description: The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) |
Ref | Expression |
---|---|
xrsp1 | ⊢ +∞ = (1.‘ℝ*𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrsex 20797 | . . 3 ⊢ ℝ*𝑠 ∈ V | |
2 | xrsbas 20798 | . . . 4 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
3 | eqid 2736 | . . . 4 ⊢ (lub‘ℝ*𝑠) = (lub‘ℝ*𝑠) | |
4 | eqid 2736 | . . . 4 ⊢ (1.‘ℝ*𝑠) = (1.‘ℝ*𝑠) | |
5 | 2, 3, 4 | p1val 18309 | . . 3 ⊢ (ℝ*𝑠 ∈ V → (1.‘ℝ*𝑠) = ((lub‘ℝ*𝑠)‘ℝ*)) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (1.‘ℝ*𝑠) = ((lub‘ℝ*𝑠)‘ℝ*) |
7 | ssid 3964 | . . 3 ⊢ ℝ* ⊆ ℝ* | |
8 | xrslt 31750 | . . . 4 ⊢ < = (lt‘ℝ*𝑠) | |
9 | xrstos 31753 | . . . . 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 | toslub 31716 | . . 3 ⊢ (ℝ* ⊆ ℝ* → ((lub‘ℝ*𝑠)‘ℝ*) = sup(ℝ*, ℝ*, < )) |
13 | 7, 12 | ax-mp 5 | . 2 ⊢ ((lub‘ℝ*𝑠)‘ℝ*) = sup(ℝ*, ℝ*, < ) |
14 | xrsup 13765 | . 2 ⊢ sup(ℝ*, ℝ*, < ) = +∞ | |
15 | 6, 13, 14 | 3eqtrri 2769 | 1 ⊢ +∞ = (1.‘ℝ*𝑠) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 ‘cfv 6493 supcsup 9372 +∞cpnf 11182 ℝ*cxr 11184 < clt 11185 ℝ*𝑠cxrs 17374 lubclub 18190 Tosetctos 18297 1.cp1 18305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9374 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-z 12496 df-dec 12615 df-uz 12760 df-fz 13417 df-struct 17011 df-slot 17046 df-ndx 17058 df-base 17076 df-plusg 17138 df-mulr 17139 df-tset 17144 df-ple 17145 df-ds 17147 df-xrs 17376 df-proset 18176 df-poset 18194 df-plt 18211 df-lub 18227 df-toset 18298 df-p1 18307 |
This theorem is referenced by: (None) |
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