infxrpnf2 - Mathbox for Glauco Siliprandi

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Theorem infxrpnf2 45457

Description: Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Assertion**
Ref	Expression
infxrpnf2	⊢ (𝐴 ⊆ ℝ^* → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))

**Proof of Theorem infxrpnf2**
Step	Hyp	Ref	Expression
1		ssdifss 4120	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 ∖ {+∞}) ⊆ ℝ^*)
2		infxrpnf 45440	. . . . 5 ⊢ ((𝐴 ∖ {+∞}) ⊆ ℝ^* → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
3	1, 2	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
5		difsnid 4791	. . . . 5 ⊢ (+∞ ∈ 𝐴 → ((𝐴 ∖ {+∞}) ∪ {+∞}) = 𝐴)
6	5	infeq1d 9495	. . . 4 ⊢ (+∞ ∈ 𝐴 → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
7	6	adantl 481	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
8	4, 7	eqtr3d 2773	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
9		difsn 4779	. . . 4 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {+∞}) = 𝐴)
10	9	infeq1d 9495	. . 3 ⊢ (¬ +∞ ∈ 𝐴 → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
11	10	adantl 481	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ ¬ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
12	8, 11	pm2.61dan 812	1 ⊢ (𝐴 ⊆ ℝ^* → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3928 ∪ cun 3929 ⊆ wss 3931 {csn 4606 infcinf 9458 +∞cpnf 11271 ℝ^*cxr 11273 < clt 11274

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-po 5566 df-so 5567 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474

This theorem is referenced by: supminfxr2 45463

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