infxrpnf2 - Mathbox for Glauco Siliprandi

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Theorem infxrpnf2 41731

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 4112	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 ∖ {+∞}) ⊆ ℝ^*)
2		infxrpnf 41713	. . . . 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 483	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf((𝐴 ∖ {+∞}), ℝ^, < ))
5		difsnid 4737	. . . . 5 ⊢ (+∞ ∈ 𝐴 → ((𝐴 ∖ {+∞}) ∪ {+∞}) = 𝐴)
6	5	infeq1d 8935	. . . 4 ⊢ (+∞ ∈ 𝐴 → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
7	6	adantl 484	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
8	4, 7	eqtr3d 2858	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
9		difsn 4725	. . . 4 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {+∞}) = 𝐴)
10	9	infeq1d 8935	. . 3 ⊢ (¬ +∞ ∈ 𝐴 → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
11	10	adantl 484	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ ¬ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))
12	8, 11	pm2.61dan 811	1 ⊢ (𝐴 ⊆ ℝ^* → inf((𝐴 ∖ {+∞}), ℝ^, < ) = inf(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3933 ∪ cun 3934 ⊆ wss 3936 {csn 4561 infcinf 8899 +∞cpnf 10666 ℝ^*cxr 10668 < clt 10669

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867

This theorem is referenced by: supminfxr2 41737

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