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Mirrors > Home > MPE Home > Th. List > infcvgaux1i | Structured version Visualization version GIF version |
Description: Auxiliary theorem for applications of supcvg 15820. 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 11657 | . . . . . 6 ⊢ (𝑦 ∈ 𝑋 → -𝐴 ∈ ℝ) |
4 | eleq1 2816 | . . . . . 6 ⊢ (𝑥 = -𝐴 → (𝑥 ∈ ℝ ↔ -𝐴 ∈ ℝ)) | |
5 | 3, 4 | syl5ibrcom 246 | . . . . 5 ⊢ (𝑦 ∈ 𝑋 → (𝑥 = -𝐴 → 𝑥 ∈ ℝ)) |
6 | 5 | rexlimiv 3143 | . . . 4 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = -𝐴 → 𝑥 ∈ ℝ) |
7 | 6 | abssi 4063 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} ⊆ ℝ |
8 | 1, 7 | eqsstri 4012 | . 2 ⊢ 𝑅 ⊆ ℝ |
9 | infcvg.3 | . . . . . 6 ⊢ 𝑍 ∈ 𝑋 | |
10 | eqid 2727 | . . . . . 6 ⊢ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴 | |
11 | 10 | nfth 1796 | . . . . . . 7 ⊢ Ⅎ𝑦-⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴 |
12 | csbeq1a 3903 | . . . . . . . . 9 ⊢ (𝑦 = 𝑍 → 𝐴 = ⦋𝑍 / 𝑦⦌𝐴) | |
13 | 12 | negeqd 11470 | . . . . . . . 8 ⊢ (𝑦 = 𝑍 → -𝐴 = -⦋𝑍 / 𝑦⦌𝐴) |
14 | 13 | eqeq2d 2738 | . . . . . . 7 ⊢ (𝑦 = 𝑍 → (-⦋𝑍 / 𝑦⦌𝐴 = -𝐴 ↔ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴)) |
15 | 11, 14 | rspce 3596 | . . . . . 6 ⊢ ((𝑍 ∈ 𝑋 ∧ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴) → ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴) |
16 | 9, 10, 15 | mp2an 691 | . . . . 5 ⊢ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴 |
17 | negex 11474 | . . . . . 6 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ V | |
18 | nfcsb1v 3914 | . . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑍 / 𝑦⦌𝐴 | |
19 | 18 | nfneg 11472 | . . . . . . . 8 ⊢ Ⅎ𝑦-⦋𝑍 / 𝑦⦌𝐴 |
20 | 19 | nfeq2 2915 | . . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = -⦋𝑍 / 𝑦⦌𝐴 |
21 | eqeq1 2731 | . . . . . . 7 ⊢ (𝑥 = -⦋𝑍 / 𝑦⦌𝐴 → (𝑥 = -𝐴 ↔ -⦋𝑍 / 𝑦⦌𝐴 = -𝐴)) | |
22 | 20, 21 | rexbid 3266 | . . . . . 6 ⊢ (𝑥 = -⦋𝑍 / 𝑦⦌𝐴 → (∃𝑦 ∈ 𝑋 𝑥 = -𝐴 ↔ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴)) |
23 | 17, 22 | elab 3665 | . . . . 5 ⊢ (-⦋𝑍 / 𝑦⦌𝐴 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} ↔ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴) |
24 | 16, 23 | mpbir 230 | . . . 4 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} |
25 | 24, 1 | eleqtrri 2827 | . . 3 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ 𝑅 |
26 | 25 | ne0ii 4333 | . 2 ⊢ 𝑅 ≠ ∅ |
27 | infcvg.4 | . 2 ⊢ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧 | |
28 | 8, 26, 27 | 3pm3.2i 1337 | 1 ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {cab 2704 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 ⦋csb 3889 ⊆ wss 3944 ∅c0 4318 class class class wbr 5142 ℝcr 11123 ≤ cle 11265 -cneg 11461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-ltxr 11269 df-sub 11462 df-neg 11463 |
This theorem is referenced by: infcvgaux2i 15822 |
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