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Mirrors > Home > MPE Home > Th. List > infcvgaux2i | Structured version Visualization version GIF version |
Description: Auxiliary theorem for applications of supcvg 15211. (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 2821 | . . . . . 6 ⊢ -𝐵 = -𝐵 | |
3 | infcvg.13 | . . . . . . . 8 ⊢ (𝑦 = 𝐶 → 𝐴 = 𝐵) | |
4 | 3 | negeqd 10880 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → -𝐴 = -𝐵) |
5 | 4 | rspceeqv 3638 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ -𝐵 = -𝐵) → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
6 | 2, 5 | mpan2 689 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
7 | negex 10884 | . . . . . 6 ⊢ -𝐵 ∈ V | |
8 | eqeq1 2825 | . . . . . . 7 ⊢ (𝑥 = -𝐵 → (𝑥 = -𝐴 ↔ -𝐵 = -𝐴)) | |
9 | 8 | rexbidv 3297 | . . . . . 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 3670 | . . . . 5 ⊢ (-𝐵 ∈ 𝑅 ↔ ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
12 | 6, 11 | sylibr 236 | . . . 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 15212 | . . . . 5 ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) |
17 | 16 | suprubii 11616 | . . . 4 ⊢ (-𝐵 ∈ 𝑅 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
18 | 12, 17 | syl 17 | . . 3 ⊢ (𝐶 ∈ 𝑋 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
19 | 3 | eleq1d 2897 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
20 | 19, 13 | vtoclga 3574 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → 𝐵 ∈ ℝ) |
21 | 16 | suprclii 11615 | . . . 4 ⊢ sup(𝑅, ℝ, < ) ∈ ℝ |
22 | lenegcon1 11144 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ sup(𝑅, ℝ, < ) ∈ ℝ) → (-𝐵 ≤ sup(𝑅, ℝ, < ) ↔ -sup(𝑅, ℝ, < ) ≤ 𝐵)) | |
23 | 20, 21, 22 | sylancl 588 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (-𝐵 ≤ sup(𝑅, ℝ, < ) ↔ -sup(𝑅, ℝ, < ) ≤ 𝐵)) |
24 | 18, 23 | mpbid 234 | . 2 ⊢ (𝐶 ∈ 𝑋 → -sup(𝑅, ℝ, < ) ≤ 𝐵) |
25 | 1, 24 | eqbrtrid 5101 | 1 ⊢ (𝐶 ∈ 𝑋 → 𝑆 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 supcsup 8904 ℝcr 10536 < clt 10675 ≤ cle 10676 -cneg 10871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 |
This theorem is referenced by: (None) |
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