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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > infxrlbrnmpt2 | Structured version Visualization version GIF version |
Description: A member of a nonempty indexed set of reals is greater than or equal to the set's lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
infxrlbrnmpt2.x | ⊢ Ⅎ𝑥𝜑 |
infxrlbrnmpt2.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
infxrlbrnmpt2.c | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
infxrlbrnmpt2.d | ⊢ (𝜑 → 𝐷 ∈ ℝ*) |
infxrlbrnmpt2.e | ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
infxrlbrnmpt2 | ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infxrlbrnmpt2.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | eqid 2727 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | infxrlbrnmpt2.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
4 | 1, 2, 3 | rnmptssd 44482 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*) |
5 | infxrlbrnmpt2.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
6 | infxrlbrnmpt2.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ*) | |
7 | infxrlbrnmpt2.e | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
8 | 2, 7 | elrnmpt1s 5953 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ ℝ*) → 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
9 | 5, 6, 8 | syl2anc 583 | . 2 ⊢ (𝜑 → 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | infxrlb 13331 | . 2 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ* ∧ 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐷) | |
11 | 4, 9, 10 | syl2anc 583 | 1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ⊆ wss 3944 class class class wbr 5142 ↦ cmpt 5225 ran crn 5673 infcinf 9450 ℝ*cxr 11263 < clt 11264 ≤ cle 11265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-sup 9451 df-inf 9452 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 |
This theorem is referenced by: limsuplesup 45000 |
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