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Mirrors > Home > ILE Home > Th. List > caucvgsrlemasr | GIF version |
Description: Lemma for caucvgsr 7610. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.) |
Ref | Expression |
---|---|
caucvgsrlemasr.bnd | ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) |
Ref | Expression |
---|---|
caucvgsrlemasr | ⊢ (𝜑 → 𝐴 ∈ R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgsrlemasr.bnd | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚)) | |
2 | ltrelsr 7546 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
3 | 2 | brel 4591 | . . . . 5 ⊢ (𝐴 <R (𝐹‘𝑚) → (𝐴 ∈ R ∧ (𝐹‘𝑚) ∈ R)) |
4 | 3 | simpld 111 | . . . 4 ⊢ (𝐴 <R (𝐹‘𝑚) → 𝐴 ∈ R) |
5 | 4 | ralimi 2495 | . . 3 ⊢ (∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚) → ∀𝑚 ∈ N 𝐴 ∈ R) |
6 | 1, 5 | syl 14 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 ∈ R) |
7 | 1pi 7123 | . . 3 ⊢ 1o ∈ N | |
8 | elex2 2702 | . . 3 ⊢ (1o ∈ N → ∃𝑥 𝑥 ∈ N) | |
9 | r19.3rmv 3453 | . . 3 ⊢ (∃𝑥 𝑥 ∈ N → (𝐴 ∈ R ↔ ∀𝑚 ∈ N 𝐴 ∈ R)) | |
10 | 7, 8, 9 | mp2b 8 | . 2 ⊢ (𝐴 ∈ R ↔ ∀𝑚 ∈ N 𝐴 ∈ R) |
11 | 6, 10 | sylibr 133 | 1 ⊢ (𝜑 → 𝐴 ∈ R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∃wex 1468 ∈ wcel 1480 ∀wral 2416 class class class wbr 3929 ‘cfv 5123 1oc1o 6306 Ncnpi 7080 Rcnr 7105 <R cltr 7111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-suc 4293 df-iom 4505 df-xp 4545 df-1o 6313 df-ni 7112 df-ltr 7538 |
This theorem is referenced by: caucvgsrlemoffval 7604 caucvgsrlemofff 7605 caucvgsrlemoffcau 7606 caucvgsrlemoffgt1 7607 caucvgsrlemoffres 7608 |
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