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Mirrors > Home > MPE Home > Th. List > infcvgaux2i | Structured version Visualization version GIF version |
Description: Auxiliary theorem for applications of supcvg 15798. (Contributed by NM, 4-Mar-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 |
infcvg.5a | ⊢ 𝑆 = -sup(𝑅, ℝ, < ) |
infcvg.13 | ⊢ (𝑦 = 𝐶 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
infcvgaux2i | ⊢ (𝐶 ∈ 𝑋 → 𝑆 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infcvg.5a | . 2 ⊢ 𝑆 = -sup(𝑅, ℝ, < ) | |
2 | eqid 2732 | . . . . . 6 ⊢ -𝐵 = -𝐵 | |
3 | infcvg.13 | . . . . . . . 8 ⊢ (𝑦 = 𝐶 → 𝐴 = 𝐵) | |
4 | 3 | negeqd 11450 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → -𝐴 = -𝐵) |
5 | 4 | rspceeqv 3632 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ -𝐵 = -𝐵) → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
6 | 2, 5 | mpan2 689 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
7 | negex 11454 | . . . . . 6 ⊢ -𝐵 ∈ V | |
8 | eqeq1 2736 | . . . . . . 7 ⊢ (𝑥 = -𝐵 → (𝑥 = -𝐴 ↔ -𝐵 = -𝐴)) | |
9 | 8 | rexbidv 3178 | . . . . . 6 ⊢ (𝑥 = -𝐵 → (∃𝑦 ∈ 𝑋 𝑥 = -𝐴 ↔ ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴)) |
10 | infcvg.1 | . . . . . 6 ⊢ 𝑅 = {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} | |
11 | 7, 9, 10 | elab2 3671 | . . . . 5 ⊢ (-𝐵 ∈ 𝑅 ↔ ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
12 | 6, 11 | sylibr 233 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → -𝐵 ∈ 𝑅) |
13 | infcvg.2 | . . . . . 6 ⊢ (𝑦 ∈ 𝑋 → 𝐴 ∈ ℝ) | |
14 | infcvg.3 | . . . . . 6 ⊢ 𝑍 ∈ 𝑋 | |
15 | infcvg.4 | . . . . . 6 ⊢ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧 | |
16 | 10, 13, 14, 15 | infcvgaux1i 15799 | . . . . 5 ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) |
17 | 16 | suprubii 12185 | . . . 4 ⊢ (-𝐵 ∈ 𝑅 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
18 | 12, 17 | syl 17 | . . 3 ⊢ (𝐶 ∈ 𝑋 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
19 | 3 | eleq1d 2818 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
20 | 19, 13 | vtoclga 3565 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → 𝐵 ∈ ℝ) |
21 | 16 | suprclii 12184 | . . . 4 ⊢ sup(𝑅, ℝ, < ) ∈ ℝ |
22 | lenegcon1 11714 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ sup(𝑅, ℝ, < ) ∈ ℝ) → (-𝐵 ≤ sup(𝑅, ℝ, < ) ↔ -sup(𝑅, ℝ, < ) ≤ 𝐵)) | |
23 | 20, 21, 22 | sylancl 586 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (-𝐵 ≤ sup(𝑅, ℝ, < ) ↔ -sup(𝑅, ℝ, < ) ≤ 𝐵)) |
24 | 18, 23 | mpbid 231 | . 2 ⊢ (𝐶 ∈ 𝑋 → -sup(𝑅, ℝ, < ) ≤ 𝐵) |
25 | 1, 24 | eqbrtrid 5182 | 1 ⊢ (𝐶 ∈ 𝑋 → 𝑆 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 {cab 2709 ∀wral 3061 ∃wrex 3070 class class class wbr 5147 supcsup 9431 ℝcr 11105 < clt 11244 ≤ cle 11245 -cneg 11441 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 |
This theorem is referenced by: (None) |
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