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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 4117 | . . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴) | |
2 | 1 | supeq1i 9391 | . 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(({-∞} ∪ 𝐴), ℝ*, < ) |
3 | mnfxr 11220 | . . . 4 ⊢ -∞ ∈ ℝ* | |
4 | snssi 4772 | . . . 4 ⊢ (-∞ ∈ ℝ* → {-∞} ⊆ ℝ*) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ (𝐴 ⊆ ℝ* → {-∞} ⊆ ℝ*) |
6 | id 22 | . . 3 ⊢ (𝐴 ⊆ ℝ* → 𝐴 ⊆ ℝ*) | |
7 | xrltso 13069 | . . . . 5 ⊢ < Or ℝ* | |
8 | supsn 9416 | . . . . 5 ⊢ (( < Or ℝ* ∧ -∞ ∈ ℝ*) → sup({-∞}, ℝ*, < ) = -∞) | |
9 | 7, 3, 8 | mp2an 691 | . . . 4 ⊢ sup({-∞}, ℝ*, < ) = -∞ |
10 | supxrcl 13243 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
11 | mnfle 13063 | . . . . 5 ⊢ (sup(𝐴, ℝ*, < ) ∈ ℝ* → -∞ ≤ sup(𝐴, ℝ*, < )) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → -∞ ≤ sup(𝐴, ℝ*, < )) |
13 | 9, 12 | eqbrtrid 5144 | . . 3 ⊢ (𝐴 ⊆ ℝ* → sup({-∞}, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) |
14 | supxrun 13244 | . . 3 ⊢ (({-∞} ⊆ ℝ* ∧ 𝐴 ⊆ ℝ* ∧ sup({-∞}, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) → sup(({-∞} ∪ 𝐴), ℝ*, < ) = sup(𝐴, ℝ*, < )) | |
15 | 5, 6, 13, 14 | syl3anc 1372 | . 2 ⊢ (𝐴 ⊆ ℝ* → sup(({-∞} ∪ 𝐴), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
16 | 2, 15 | eqtrid 2785 | 1 ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cun 3912 ⊆ wss 3914 {csn 4590 class class class wbr 5109 Or wor 5548 supcsup 9384 -∞cmnf 11195 ℝ*cxr 11196 < clt 11197 ≤ cle 11198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 |
This theorem is referenced by: supxrmnf2 43758 |
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