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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > suprubrnmpt2 | Structured version Visualization version GIF version |
Description: A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
suprubrnmpt2.x | ⊢ Ⅎ𝑥𝜑 |
suprubrnmpt2.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
suprubrnmpt2.l | ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
suprubrnmpt2.c | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
suprubrnmpt2.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
suprubrnmpt2.i | ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
suprubrnmpt2 | ⊢ (𝜑 → 𝐷 ≤ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprubrnmpt2.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | suprubrnmpt2.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | 1, 2, 3 | rnmptssd 43314 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ) |
5 | suprubrnmpt2.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
6 | suprubrnmpt2.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | suprubrnmpt2.i | . . . . 5 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
8 | 2, 7 | elrnmpt1s 5910 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ ℝ) → 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
9 | 5, 6, 8 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | 9 | ne0d 4293 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) |
11 | suprubrnmpt2.l | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) | |
12 | 1, 11 | rnmptbdd 43370 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦) |
13 | 4, 10, 12, 9 | suprubd 12075 | 1 ⊢ (𝜑 → 𝐷 ≤ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ∀wral 3062 ∃wrex 3071 class class class wbr 5103 ↦ cmpt 5186 ran crn 5632 supcsup 9334 ℝcr 11008 < clt 11147 ≤ cle 11148 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 |
This theorem is referenced by: (None) |
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