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| Description: Auxiliary theorem for applications of supcvg 15892. Hypothesis for several supremum theorems. (Contributed by NM, 8-Feb-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 | 
| Ref | Expression | 
|---|---|
| infcvgaux1i | ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | infcvg.1 | . . 3 ⊢ 𝑅 = {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} | |
| 2 | infcvg.2 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → 𝐴 ∈ ℝ) | |
| 3 | 2 | renegcld 11690 | . . . . . 6 ⊢ (𝑦 ∈ 𝑋 → -𝐴 ∈ ℝ) | 
| 4 | eleq1 2829 | . . . . . 6 ⊢ (𝑥 = -𝐴 → (𝑥 ∈ ℝ ↔ -𝐴 ∈ ℝ)) | |
| 5 | 3, 4 | syl5ibrcom 247 | . . . . 5 ⊢ (𝑦 ∈ 𝑋 → (𝑥 = -𝐴 → 𝑥 ∈ ℝ)) | 
| 6 | 5 | rexlimiv 3148 | . . . 4 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = -𝐴 → 𝑥 ∈ ℝ) | 
| 7 | 6 | abssi 4070 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} ⊆ ℝ | 
| 8 | 1, 7 | eqsstri 4030 | . 2 ⊢ 𝑅 ⊆ ℝ | 
| 9 | infcvg.3 | . . . . . 6 ⊢ 𝑍 ∈ 𝑋 | |
| 10 | eqid 2737 | . . . . . 6 ⊢ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴 | |
| 11 | 10 | nfth 1801 | . . . . . . 7 ⊢ Ⅎ𝑦-⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴 | 
| 12 | csbeq1a 3913 | . . . . . . . . 9 ⊢ (𝑦 = 𝑍 → 𝐴 = ⦋𝑍 / 𝑦⦌𝐴) | |
| 13 | 12 | negeqd 11502 | . . . . . . . 8 ⊢ (𝑦 = 𝑍 → -𝐴 = -⦋𝑍 / 𝑦⦌𝐴) | 
| 14 | 13 | eqeq2d 2748 | . . . . . . 7 ⊢ (𝑦 = 𝑍 → (-⦋𝑍 / 𝑦⦌𝐴 = -𝐴 ↔ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴)) | 
| 15 | 11, 14 | rspce 3611 | . . . . . 6 ⊢ ((𝑍 ∈ 𝑋 ∧ -⦋𝑍 / 𝑦⦌𝐴 = -⦋𝑍 / 𝑦⦌𝐴) → ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴) | 
| 16 | 9, 10, 15 | mp2an 692 | . . . . 5 ⊢ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴 | 
| 17 | negex 11506 | . . . . . 6 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ V | |
| 18 | nfcsb1v 3923 | . . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑍 / 𝑦⦌𝐴 | |
| 19 | 18 | nfneg 11504 | . . . . . . . 8 ⊢ Ⅎ𝑦-⦋𝑍 / 𝑦⦌𝐴 | 
| 20 | 19 | nfeq2 2923 | . . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = -⦋𝑍 / 𝑦⦌𝐴 | 
| 21 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑥 = -⦋𝑍 / 𝑦⦌𝐴 → (𝑥 = -𝐴 ↔ -⦋𝑍 / 𝑦⦌𝐴 = -𝐴)) | |
| 22 | 20, 21 | rexbid 3274 | . . . . . 6 ⊢ (𝑥 = -⦋𝑍 / 𝑦⦌𝐴 → (∃𝑦 ∈ 𝑋 𝑥 = -𝐴 ↔ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴)) | 
| 23 | 17, 22 | elab 3679 | . . . . 5 ⊢ (-⦋𝑍 / 𝑦⦌𝐴 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} ↔ ∃𝑦 ∈ 𝑋 -⦋𝑍 / 𝑦⦌𝐴 = -𝐴) | 
| 24 | 16, 23 | mpbir 231 | . . . 4 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} | 
| 25 | 24, 1 | eleqtrri 2840 | . . 3 ⊢ -⦋𝑍 / 𝑦⦌𝐴 ∈ 𝑅 | 
| 26 | 25 | ne0ii 4344 | . 2 ⊢ 𝑅 ≠ ∅ | 
| 27 | infcvg.4 | . 2 ⊢ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧 | |
| 28 | 8, 26, 27 | 3pm3.2i 1340 | 1 ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⦋csb 3899 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ℝcr 11154 ≤ cle 11296 -cneg 11493 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 | 
| This theorem is referenced by: infcvgaux2i 15894 | 
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