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Mirrors > Home > MPE Home > Th. List > supxrmnf | Structured version Visualization version GIF version |
Description: Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
supxrmnf | ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 4128 | . . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴) | |
2 | 1 | supeq1i 8905 | . 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(({-∞} ∪ 𝐴), ℝ*, < ) |
3 | mnfxr 10692 | . . . 4 ⊢ -∞ ∈ ℝ* | |
4 | snssi 4734 | . . . 4 ⊢ (-∞ ∈ ℝ* → {-∞} ⊆ ℝ*) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ (𝐴 ⊆ ℝ* → {-∞} ⊆ ℝ*) |
6 | id 22 | . . 3 ⊢ (𝐴 ⊆ ℝ* → 𝐴 ⊆ ℝ*) | |
7 | xrltso 12528 | . . . . 5 ⊢ < Or ℝ* | |
8 | supsn 8930 | . . . . 5 ⊢ (( < Or ℝ* ∧ -∞ ∈ ℝ*) → sup({-∞}, ℝ*, < ) = -∞) | |
9 | 7, 3, 8 | mp2an 690 | . . . 4 ⊢ sup({-∞}, ℝ*, < ) = -∞ |
10 | supxrcl 12702 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
11 | mnfle 12523 | . . . . 5 ⊢ (sup(𝐴, ℝ*, < ) ∈ ℝ* → -∞ ≤ sup(𝐴, ℝ*, < )) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → -∞ ≤ sup(𝐴, ℝ*, < )) |
13 | 9, 12 | eqbrtrid 5093 | . . 3 ⊢ (𝐴 ⊆ ℝ* → sup({-∞}, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) |
14 | supxrun 12703 | . . 3 ⊢ (({-∞} ⊆ ℝ* ∧ 𝐴 ⊆ ℝ* ∧ sup({-∞}, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) → sup(({-∞} ∪ 𝐴), ℝ*, < ) = sup(𝐴, ℝ*, < )) | |
15 | 5, 6, 13, 14 | syl3anc 1367 | . 2 ⊢ (𝐴 ⊆ ℝ* → sup(({-∞} ∪ 𝐴), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
16 | 2, 15 | syl5eq 2868 | 1 ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 ⊆ wss 3935 {csn 4560 class class class wbr 5058 Or wor 5467 supcsup 8898 -∞cmnf 10667 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 |
This theorem is referenced by: supxrmnf2 41700 |
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