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Mirrors > Home > MPE Home > Th. List > infcvgaux2i | Structured version Visualization version GIF version |
Description: Auxiliary theorem for applications of supcvg 15829. (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 2725 | . . . . . 6 ⊢ -𝐵 = -𝐵 | |
3 | infcvg.13 | . . . . . . . 8 ⊢ (𝑦 = 𝐶 → 𝐴 = 𝐵) | |
4 | 3 | negeqd 11479 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → -𝐴 = -𝐵) |
5 | 4 | rspceeqv 3625 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ -𝐵 = -𝐵) → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
6 | 2, 5 | mpan2 689 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
7 | negex 11483 | . . . . . 6 ⊢ -𝐵 ∈ V | |
8 | eqeq1 2729 | . . . . . . 7 ⊢ (𝑥 = -𝐵 → (𝑥 = -𝐴 ↔ -𝐵 = -𝐴)) | |
9 | 8 | rexbidv 3169 | . . . . . 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 3665 | . . . . 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 15830 | . . . . 5 ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) |
17 | 16 | suprubii 12214 | . . . 4 ⊢ (-𝐵 ∈ 𝑅 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
18 | 12, 17 | syl 17 | . . 3 ⊢ (𝐶 ∈ 𝑋 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
19 | 3 | eleq1d 2810 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
20 | 19, 13 | vtoclga 3557 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → 𝐵 ∈ ℝ) |
21 | 16 | suprclii 12213 | . . . 4 ⊢ sup(𝑅, ℝ, < ) ∈ ℝ |
22 | lenegcon1 11743 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ sup(𝑅, ℝ, < ) ∈ ℝ) → (-𝐵 ≤ sup(𝑅, ℝ, < ) ↔ -sup(𝑅, ℝ, < ) ≤ 𝐵)) | |
23 | 20, 21, 22 | sylancl 584 | . . 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 5179 | 1 ⊢ (𝐶 ∈ 𝑋 → 𝑆 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {cab 2702 ∀wral 3051 ∃wrex 3060 class class class wbr 5144 supcsup 9458 ℝcr 11132 < clt 11273 ≤ cle 11274 -cneg 11470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 |
This theorem is referenced by: (None) |
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