Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > infleinf2 | Structured version Visualization version GIF version |
Description: If any element in 𝐵 is larger or equal to an element in 𝐴, then the infimum of 𝐴 is smaller or equal to the infimum of 𝐵. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
infleinf2.x | ⊢ Ⅎ𝑥𝜑 |
infleinf2.p | ⊢ Ⅎ𝑦𝜑 |
infleinf2.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
infleinf2.b | ⊢ (𝜑 → 𝐵 ⊆ ℝ*) |
infleinf2.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
Ref | Expression |
---|---|
infleinf2 | ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infleinf2.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | infleinf2.y | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | |
3 | infleinf2.p | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
4 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐵 | |
5 | 3, 4 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐵) |
6 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑦inf(𝐴, ℝ*, < ) ≤ 𝑥 | |
7 | infleinf2.a | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | |
8 | 7 | infxrcld 39612 | . . . . . . . . 9 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
9 | 8 | 3ad2ant1 1082 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
10 | 9 | 3adant1r 1319 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
11 | 7 | sselda 3603 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ*) |
12 | 11 | 3adant3 1081 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*) |
13 | 12 | 3adant1r 1319 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*) |
14 | infleinf2.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ ℝ*) | |
15 | 14 | sselda 3603 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ ℝ*) |
16 | 15 | 3ad2ant1 1082 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → 𝑥 ∈ ℝ*) |
17 | 7 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
18 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
19 | infxrlb 12164 | . . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑦 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑦) | |
20 | 17, 18, 19 | syl2anc 693 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑦) |
21 | 20 | 3adant3 1081 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ≤ 𝑦) |
22 | 21 | 3adant1r 1319 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ≤ 𝑦) |
23 | simp3 1063 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑥) | |
24 | 10, 13, 16, 22, 23 | xrletrd 11993 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ≤ 𝑥) |
25 | 24 | 3exp 1264 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ 𝐴 → (𝑦 ≤ 𝑥 → inf(𝐴, ℝ*, < ) ≤ 𝑥))) |
26 | 5, 6, 25 | rexlimd 3026 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → inf(𝐴, ℝ*, < ) ≤ 𝑥)) |
27 | 2, 26 | mpd 15 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → inf(𝐴, ℝ*, < ) ≤ 𝑥) |
28 | 1, 27 | ralrimia 39315 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 inf(𝐴, ℝ*, < ) ≤ 𝑥) |
29 | infxrgelb 12165 | . . 3 ⊢ ((𝐵 ⊆ ℝ* ∧ inf(𝐴, ℝ*, < ) ∈ ℝ*) → (inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < ) ↔ ∀𝑥 ∈ 𝐵 inf(𝐴, ℝ*, < ) ≤ 𝑥)) | |
30 | 14, 8, 29 | syl2anc 693 | . 2 ⊢ (𝜑 → (inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < ) ↔ ∀𝑥 ∈ 𝐵 inf(𝐴, ℝ*, < ) ≤ 𝑥)) |
31 | 28, 30 | mpbird 247 | 1 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 infcinf 8347 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 |
This theorem is referenced by: infrnmptle 39650 infxrpnf 39674 |
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