Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > infcvgaux1i | Structured version Visualization version GIF version |
Description: Auxiliary theorem for applications of supcvg 14427. Hypothesis for several supremum theorems. (Contributed by NM, 8-Feb-2008.) |
Ref | Expression |
---|---|
infcvg.1 | ⊢ 𝑅 = {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} |
infcvg.2 | ⊢ (𝑦 ∈ 𝑋 → 𝐴 ∈ ℝ) |
infcvg.3 | ⊢ 𝑍 ∈ 𝑋 |
infcvg.4 | ⊢ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧 |
Ref | Expression |
---|---|
infcvgaux1i | ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infcvg.1 | . . 3 ⊢ 𝑅 = {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} | |
2 | infcvg.2 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → 𝐴 ∈ ℝ) | |
3 | 2 | renegcld 10336 | . . . . . 6 ⊢ (𝑦 ∈ 𝑋 → -𝐴 ∈ ℝ) |
4 | eleq1 2676 | . . . . . 6 ⊢ (𝑥 = -𝐴 → (𝑥 ∈ ℝ ↔ -𝐴 ∈ ℝ)) | |
5 | 3, 4 | syl5ibrcom 236 | . . . . 5 ⊢ (𝑦 ∈ 𝑋 → (𝑥 = -𝐴 → 𝑥 ∈ ℝ)) |
6 | 5 | rexlimiv 3009 | . . . 4 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = -𝐴 → 𝑥 ∈ ℝ) |
7 | 6 | abssi 3640 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} ⊆ ℝ |
8 | 1, 7 | eqsstri 3598 | . 2 ⊢ 𝑅 ⊆ ℝ |
9 | infcvg.3 | . . . . . 6 ⊢ 𝑍 ∈ 𝑋 | |
10 | eqid 2610 | . . . . . 6 ⊢ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴 | |
11 | 10 | nfth 1718 | . . . . . . 7 ⊢ Ⅎ𝑦-⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴 |
12 | csbeq1a 3508 | . . . . . . . . 9 ⊢ (𝑦 = 𝑍 → 𝐴 = ⦋𝑍 / 𝑦⦌𝐴) | |
13 | 12 | negeqd 10154 | . . . . . . . 8 ⊢ (𝑦 = 𝑍 → -𝐴 = -⦋𝑍 / 𝑦⦌𝐴) |
14 | 13 | eqeq2d 2620 | . . . . . . 7 ⊢ (𝑦 = 𝑍 → (-⦋𝑍 / 𝑦⦌𝐴 = -𝐴 ↔ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴)) |
15 | 11, 14 | rspce 3277 | . . . . . 6 ⊢ ((𝑍 ∈ 𝑋 ∧ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴) → ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴) |
16 | 9, 10, 15 | mp2an 704 | . . . . 5 ⊢ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴 |
17 | negex 10158 | . . . . . 6 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ V | |
18 | nfcsb1v 3515 | . . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑍 / 𝑦⦌𝐴 | |
19 | 18 | nfneg 10156 | . . . . . . . 8 ⊢ Ⅎ𝑦-⦋𝑍 / 𝑦⦌𝐴 |
20 | 19 | nfeq2 2766 | . . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = -⦋𝑍 / 𝑦⦌𝐴 |
21 | eqeq1 2614 | . . . . . . 7 ⊢ (𝑥 = -⦋𝑍 / 𝑦⦌𝐴 → (𝑥 = -𝐴 ↔ -⦋𝑍 / 𝑦⦌𝐴 = -𝐴)) | |
22 | 20, 21 | rexbid 3033 | . . . . . 6 ⊢ (𝑥 = -⦋𝑍 / 𝑦⦌𝐴 → (∃𝑦 ∈ 𝑋 𝑥 = -𝐴 ↔ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴)) |
23 | 17, 22 | elab 3319 | . . . . 5 ⊢ (-⦋𝑍 / 𝑦⦌𝐴 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} ↔ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴) |
24 | 16, 23 | mpbir 220 | . . . 4 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} |
25 | 24, 1 | eleqtrri 2687 | . . 3 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ 𝑅 |
26 | 25 | ne0ii 3882 | . 2 ⊢ 𝑅 ≠ ∅ |
27 | infcvg.4 | . 2 ⊢ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧 | |
28 | 8, 26, 27 | 3pm3.2i 1232 | 1 ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⦋csb 3499 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ℝcr 9814 ≤ cle 9954 -cneg 10146 |
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-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 |
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-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 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-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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 |
This theorem is referenced by: infcvgaux2i 14429 |
Copyright terms: Public domain | W3C validator |