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Mirrors > Home > MPE Home > Th. List > infcvgaux2i | Structured version Visualization version GIF version |
Description: Auxiliary theorem for applications of supcvg 15820. (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 2727 | . . . . . 6 ⊢ -𝐵 = -𝐵 | |
3 | infcvg.13 | . . . . . . . 8 ⊢ (𝑦 = 𝐶 → 𝐴 = 𝐵) | |
4 | 3 | negeqd 11470 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → -𝐴 = -𝐵) |
5 | 4 | rspceeqv 3629 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ -𝐵 = -𝐵) → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
6 | 2, 5 | mpan2 690 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 -𝐵 = -𝐴) |
7 | negex 11474 | . . . . . 6 ⊢ -𝐵 ∈ V | |
8 | eqeq1 2731 | . . . . . . 7 ⊢ (𝑥 = -𝐵 → (𝑥 = -𝐴 ↔ -𝐵 = -𝐴)) | |
9 | 8 | rexbidv 3173 | . . . . . 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 3669 | . . . . 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 15821 | . . . . 5 ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) |
17 | 16 | suprubii 12205 | . . . 4 ⊢ (-𝐵 ∈ 𝑅 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
18 | 12, 17 | syl 17 | . . 3 ⊢ (𝐶 ∈ 𝑋 → -𝐵 ≤ sup(𝑅, ℝ, < )) |
19 | 3 | eleq1d 2813 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
20 | 19, 13 | vtoclga 3561 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → 𝐵 ∈ ℝ) |
21 | 16 | suprclii 12204 | . . . 4 ⊢ sup(𝑅, ℝ, < ) ∈ ℝ |
22 | lenegcon1 11734 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ sup(𝑅, ℝ, < ) ∈ ℝ) → (-𝐵 ≤ sup(𝑅, ℝ, < ) ↔ -sup(𝑅, ℝ, < ) ≤ 𝐵)) | |
23 | 20, 21, 22 | sylancl 585 | . . 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 5177 | 1 ⊢ (𝐶 ∈ 𝑋 → 𝑆 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {cab 2704 ∀wral 3056 ∃wrex 3065 class class class wbr 5142 supcsup 9449 ℝcr 11123 < clt 11264 ≤ 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 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
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-rmo 3371 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-sup 9451 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 |
This theorem is referenced by: (None) |
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