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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > infxrpnf2 | Structured version Visualization version GIF version |
Description: Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
infxrpnf2 | ⊢ (𝐴 ⊆ ℝ* → inf((𝐴 ∖ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdifss 4128 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ∖ {+∞}) ⊆ ℝ*) | |
2 | infxrpnf 44891 | . . . . 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 479 | . . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ*, < ) = inf((𝐴 ∖ {+∞}), ℝ*, < )) |
5 | difsnid 4809 | . . . . 5 ⊢ (+∞ ∈ 𝐴 → ((𝐴 ∖ {+∞}) ∪ {+∞}) = 𝐴) | |
6 | 5 | infeq1d 9500 | . . . 4 ⊢ (+∞ ∈ 𝐴 → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) |
7 | 6 | adantl 480 | . . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → inf(((𝐴 ∖ {+∞}) ∪ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) |
8 | 4, 7 | eqtr3d 2767 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → inf((𝐴 ∖ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) |
9 | difsn 4797 | . . . 4 ⊢ (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {+∞}) = 𝐴) | |
10 | 9 | infeq1d 9500 | . . 3 ⊢ (¬ +∞ ∈ 𝐴 → inf((𝐴 ∖ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) |
11 | 10 | adantl 480 | . 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 394 = wceq 1533 ∈ wcel 2098 ∖ cdif 3936 ∪ cun 3937 ⊆ wss 3939 {csn 4624 infcinf 9464 +∞cpnf 11275 ℝ*cxr 11277 < clt 11278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 |
This theorem is referenced by: supminfxr2 44914 |
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