| 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 greater than or equal to an element in 𝐴, then the infimum of 𝐴 is less than 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 1914 | . . . . . 6 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐵 | |
| 5 | 3, 4 | nfan 1899 | . . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐵) |
| 6 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦inf(𝐴, ℝ*, < ) ≤ 𝑥 | |
| 7 | infleinf2.a | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | |
| 8 | 7 | infxrcld 45400 | . . . . . . . . 9 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
| 9 | 8 | 3ad2ant1 1134 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
| 10 | 9 | 3adant1r 1178 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
| 11 | 7 | sselda 3983 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ*) |
| 12 | 11 | 3adant3 1133 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*) |
| 13 | 12 | 3adant1r 1178 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → 𝑦 ∈ ℝ*) |
| 14 | infleinf2.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ ℝ*) | |
| 15 | 14 | sselda 3983 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ ℝ*) |
| 16 | 15 | 3ad2ant1 1134 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → 𝑥 ∈ ℝ*) |
| 17 | 7 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
| 18 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
| 19 | infxrlb 13376 | . . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑦 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑦) | |
| 20 | 17, 18, 19 | syl2anc 584 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑦) |
| 21 | 20 | 3adant3 1133 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ≤ 𝑦) |
| 22 | 21 | 3adant1r 1178 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ≤ 𝑦) |
| 23 | simp3 1139 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑥) | |
| 24 | 10, 13, 16, 22, 23 | xrletrd 13204 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦 ≤ 𝑥) → inf(𝐴, ℝ*, < ) ≤ 𝑥) |
| 25 | 24 | 3exp 1120 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑦 ∈ 𝐴 → (𝑦 ≤ 𝑥 → inf(𝐴, ℝ*, < ) ≤ 𝑥))) |
| 26 | 5, 6, 25 | rexlimd 3266 | . . . 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 3258 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 inf(𝐴, ℝ*, < ) ≤ 𝑥) |
| 29 | infxrgelb 13377 | . . 3 ⊢ ((𝐵 ⊆ ℝ* ∧ inf(𝐴, ℝ*, < ) ∈ ℝ*) → (inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < ) ↔ ∀𝑥 ∈ 𝐵 inf(𝐴, ℝ*, < ) ≤ 𝑥)) | |
| 30 | 14, 8, 29 | syl2anc 584 | . 2 ⊢ (𝜑 → (inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < ) ↔ ∀𝑥 ∈ 𝐵 inf(𝐴, ℝ*, < ) ≤ 𝑥)) |
| 31 | 28, 30 | mpbird 257 | 1 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 infcinf 9481 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: infrnmptle 45434 infxrpnf 45457 |
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