Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rlimle | Structured version Visualization version GIF version |
Description: Comparison of the limits of two sequences. (Contributed by Mario Carneiro, 10-May-2016.) |
Ref | Expression |
---|---|
rlimle.1 | ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) |
rlimle.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷) |
rlimle.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸) |
rlimle.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
rlimle.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
rlimle.6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
rlimle | ⊢ (𝜑 → 𝐷 ≤ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimle.1 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) | |
2 | rlimle.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) | |
3 | rlimle.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | rlimle.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸) | |
5 | rlimle.2 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷) | |
6 | 2, 3, 4, 5 | rlimsub 14222 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐵)) ⇝𝑟 (𝐸 − 𝐷)) |
7 | 2, 3 | resubcld 10337 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 − 𝐵) ∈ ℝ) |
8 | rlimle.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶) | |
9 | 2, 3 | subge0d 10496 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ (𝐶 − 𝐵) ↔ 𝐵 ≤ 𝐶)) |
10 | 8, 9 | mpbird 246 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (𝐶 − 𝐵)) |
11 | 1, 6, 7, 10 | rlimge0 14160 | . 2 ⊢ (𝜑 → 0 ≤ (𝐸 − 𝐷)) |
12 | 1, 4, 2 | rlimrecl 14159 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
13 | 1, 5, 3 | rlimrecl 14159 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
14 | 12, 13 | subge0d 10496 | . 2 ⊢ (𝜑 → (0 ≤ (𝐸 − 𝐷) ↔ 𝐷 ≤ 𝐸)) |
15 | 11, 14 | mpbid 221 | 1 ⊢ (𝜑 → 𝐷 ≤ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ↦ cmpt 4643 (class class class)co 6549 supcsup 8229 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 ⇝𝑟 crli 14064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-rlim 14068 |
This theorem is referenced by: dvfsumrlimge0 23597 dvfsumrlim2 23599 |
Copyright terms: Public domain | W3C validator |