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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 4126 | . . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴) | |
2 | 1 | supeq1i 8899 | . 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(({-∞} ∪ 𝐴), ℝ*, < ) |
3 | mnfxr 10686 | . . . 4 ⊢ -∞ ∈ ℝ* | |
4 | snssi 4733 | . . . 4 ⊢ (-∞ ∈ ℝ* → {-∞} ⊆ ℝ*) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ (𝐴 ⊆ ℝ* → {-∞} ⊆ ℝ*) |
6 | id 22 | . . 3 ⊢ (𝐴 ⊆ ℝ* → 𝐴 ⊆ ℝ*) | |
7 | xrltso 12522 | . . . . 5 ⊢ < Or ℝ* | |
8 | supsn 8924 | . . . . 5 ⊢ (( < Or ℝ* ∧ -∞ ∈ ℝ*) → sup({-∞}, ℝ*, < ) = -∞) | |
9 | 7, 3, 8 | mp2an 688 | . . . 4 ⊢ sup({-∞}, ℝ*, < ) = -∞ |
10 | supxrcl 12696 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
11 | mnfle 12517 | . . . . 5 ⊢ (sup(𝐴, ℝ*, < ) ∈ ℝ* → -∞ ≤ sup(𝐴, ℝ*, < )) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → -∞ ≤ sup(𝐴, ℝ*, < )) |
13 | 9, 12 | eqbrtrid 5092 | . . 3 ⊢ (𝐴 ⊆ ℝ* → sup({-∞}, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) |
14 | supxrun 12697 | . . 3 ⊢ (({-∞} ⊆ ℝ* ∧ 𝐴 ⊆ ℝ* ∧ sup({-∞}, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) → sup(({-∞} ∪ 𝐴), ℝ*, < ) = sup(𝐴, ℝ*, < )) | |
15 | 5, 6, 13, 14 | syl3anc 1363 | . 2 ⊢ (𝐴 ⊆ ℝ* → sup(({-∞} ∪ 𝐴), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
16 | 2, 15 | syl5eq 2865 | 1 ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 ⊆ wss 3933 {csn 4557 class class class wbr 5057 Or wor 5466 supcsup 8892 -∞cmnf 10661 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 |
This theorem is referenced by: supxrmnf2 41583 |
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