supxrmnf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > supxrmnf		Structured version Visualization version GIF version

Theorem supxrmnf 13344

Description: Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)

**Assertion**
Ref	Expression
supxrmnf	⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

**Proof of Theorem supxrmnf**
Step	Hyp	Ref	Expression
1		uncom 4113	. . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴)
2	1	supeq1i 9408	. 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(({-∞} ∪ 𝐴), ℝ^, < )
3		mnfxr 11267	. . . 4 ⊢ -∞ ∈ ℝ^*
4		snssi 4752	. . . 4 ⊢ (-∞ ∈ ℝ^* → {-∞} ⊆ ℝ^*)
5	3, 4	mp1i 14	. . 3 ⊢ (𝐴 ⊆ ℝ^* → {-∞} ⊆ ℝ^*)
6		id 23	. . 3 ⊢ (𝐴 ⊆ ℝ^* → 𝐴 ⊆ ℝ^*)
7		xrltso 13167	. . . . 5 ⊢ < Or ℝ^*
8		supsn 9434	. . . . 5 ⊢ (( < Or ℝ^* ∧ -∞ ∈ ℝ^) → sup({-∞}, ℝ^, < ) = -∞)
9	7, 3, 8	mp2an 704	. . . 4 ⊢ sup({-∞}, ℝ^*, < ) = -∞
10		supxrcl 13342	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
11		mnfle 13161	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → -∞ ≤ sup(𝐴, ℝ^*, < ))
12	10, 11	syl 18	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → -∞ ≤ sup(𝐴, ℝ^*, < ))
13	9, 12	eqbrtrid 5147	. . 3 ⊢ (𝐴 ⊆ ℝ^* → sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < ))
14		supxrun 13343	. . 3 ⊢ (({-∞} ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^* ∧ sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < )) → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
15	5, 6, 13, 14	syl3anc 1398	. 2 ⊢ (𝐴 ⊆ ℝ^* → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
16	2, 15	eqtrid 2810	1 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1570 ∈ wcel 2143 ∪ cun 3904 ⊆ wss 3906 {csn 4590 class class class wbr 5110 Or wor 5570 supcsup 9401 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-sup 9403 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445

This theorem is referenced by: supxrmnf2 46130

W3C validator