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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 2798 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | infxrlbrnmpt2.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
4 | 1, 2, 3 | rnmptssd 41824 | . 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 5793 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ ℝ*) → 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
9 | 5, 6, 8 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | infxrlb 12715 | . 2 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ* ∧ 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐷) | |
11 | 4, 9, 10 | syl2anc 587 | 1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 infcinf 8889 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 |
This theorem is referenced by: limsuplesup 42341 |
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