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Mirrors > Home > MPE Home > Th. List > infcvgaux2i | Structured version Visualization version GIF version |
Description: Auxiliary theorem for applications of supcvg 15420. (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 2737 | . . . . . 6 ⊢ -𝐵 = -𝐵 | |
3 | infcvg.13 | . . . . . . . 8 ⊢ (𝑦 = 𝐶 → 𝐴 = 𝐵) | |
4 | 3 | negeqd 11072 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → -𝐴 = -𝐵) |
5 | 4 | rspceeqv 3552 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ -𝐵 = -𝐵) → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
6 | 2, 5 | mpan2 691 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
7 | negex 11076 | . . . . . 6 ⊢ -𝐵 ∈ V | |
8 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑥 = -𝐵 → (𝑥 = -𝐴 ↔ -𝐵 = -𝐴)) | |
9 | 8 | rexbidv 3216 | . . . . . 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 3591 | . . . . 5 ⊢ (-𝐵 ∈ 𝑅 ↔ ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
12 | 6, 11 | sylibr 237 | . . . 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 15421 | . . . . 5 ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) |
17 | 16 | suprubii 11807 | . . . 4 ⊢ (-𝐵 ∈ 𝑅 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
18 | 12, 17 | syl 17 | . . 3 ⊢ (𝐶 ∈ 𝑋 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
19 | 3 | eleq1d 2822 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
20 | 19, 13 | vtoclga 3489 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → 𝐵 ∈ ℝ) |
21 | 16 | suprclii 11806 | . . . 4 ⊢ sup(𝑅, ℝ, < ) ∈ ℝ |
22 | lenegcon1 11336 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ sup(𝑅, ℝ, < ) ∈ ℝ) → (-𝐵 ≤ sup(𝑅, ℝ, < ) ↔ -sup(𝑅, ℝ, < ) ≤ 𝐵)) | |
23 | 20, 21, 22 | sylancl 589 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (-𝐵 ≤ sup(𝑅, ℝ, < ) ↔ -sup(𝑅, ℝ, < ) ≤ 𝐵)) |
24 | 18, 23 | mpbid 235 | . 2 ⊢ (𝐶 ∈ 𝑋 → -sup(𝑅, ℝ, < ) ≤ 𝐵) |
25 | 1, 24 | eqbrtrid 5088 | 1 ⊢ (𝐶 ∈ 𝑋 → 𝑆 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 {cab 2714 ∀wral 3061 ∃wrex 3062 class class class wbr 5053 supcsup 9056 ℝcr 10728 < clt 10867 ≤ cle 10868 -cneg 11063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 |
This theorem is referenced by: (None) |
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