|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 4157 | . . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴) | |
| 2 | 1 | supeq1i 9488 | . 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(({-∞} ∪ 𝐴), ℝ*, < ) | 
| 3 | mnfxr 11319 | . . . 4 ⊢ -∞ ∈ ℝ* | |
| 4 | snssi 4807 | . . . 4 ⊢ (-∞ ∈ ℝ* → {-∞} ⊆ ℝ*) | |
| 5 | 3, 4 | mp1i 13 | . . 3 ⊢ (𝐴 ⊆ ℝ* → {-∞} ⊆ ℝ*) | 
| 6 | id 22 | . . 3 ⊢ (𝐴 ⊆ ℝ* → 𝐴 ⊆ ℝ*) | |
| 7 | xrltso 13184 | . . . . 5 ⊢ < Or ℝ* | |
| 8 | supsn 9513 | . . . . 5 ⊢ (( < Or ℝ* ∧ -∞ ∈ ℝ*) → sup({-∞}, ℝ*, < ) = -∞) | |
| 9 | 7, 3, 8 | mp2an 692 | . . . 4 ⊢ sup({-∞}, ℝ*, < ) = -∞ | 
| 10 | supxrcl 13358 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
| 11 | mnfle 13178 | . . . . 5 ⊢ (sup(𝐴, ℝ*, < ) ∈ ℝ* → -∞ ≤ sup(𝐴, ℝ*, < )) | |
| 12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → -∞ ≤ sup(𝐴, ℝ*, < )) | 
| 13 | 9, 12 | eqbrtrid 5177 | . . 3 ⊢ (𝐴 ⊆ ℝ* → sup({-∞}, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) | 
| 14 | supxrun 13359 | . . 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 2788 | 1 ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∪ cun 3948 ⊆ wss 3950 {csn 4625 class class class wbr 5142 Or wor 5590 supcsup 9481 -∞cmnf 11294 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 | 
| This theorem is referenced by: supxrmnf2 45449 | 
| Copyright terms: Public domain | W3C validator |