Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 2734 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | suprubrnmpt2.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | 1, 2, 3 | rnmptssd 42360 | . 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 5815 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ ℝ) → 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
9 | 5, 6, 8 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | 9 | ne0d 4240 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) |
11 | suprubrnmpt2.l | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) | |
12 | 1, 11 | rnmptbdd 42414 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦) |
13 | 4, 10, 12, 9 | suprubd 11777 | 1 ⊢ (𝜑 → 𝐷 ≤ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 ∀wral 3054 ∃wrex 3055 class class class wbr 5043 ↦ cmpt 5124 ran crn 5541 supcsup 9045 ℝcr 10711 < clt 10850 ≤ cle 10851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |