![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > caucvgsrlemasr | GIF version |
Description: Lemma for caucvgsr 7345. 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 7282 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
3 | 2 | brel 4490 | . . . . 5 ⊢ (𝐴 <R (𝐹‘𝑚) → (𝐴 ∈ R ∧ (𝐹‘𝑚) ∈ R)) |
4 | 3 | simpld 110 | . . . 4 ⊢ (𝐴 <R (𝐹‘𝑚) → 𝐴 ∈ R) |
5 | 4 | ralimi 2438 | . . 3 ⊢ (∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚) → ∀𝑚 ∈ N 𝐴 ∈ R) |
6 | 1, 5 | syl 14 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 ∈ R) |
7 | 1pi 6872 | . . 3 ⊢ 1𝑜 ∈ N | |
8 | elex2 2635 | . . 3 ⊢ (1𝑜 ∈ N → ∃𝑥 𝑥 ∈ N) | |
9 | r19.3rmv 3372 | . . 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 132 | 1 ⊢ (𝜑 → 𝐴 ∈ R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∃wex 1426 ∈ wcel 1438 ∀wral 2359 class class class wbr 3845 ‘cfv 5015 1𝑜c1o 6174 Ncnpi 6829 Rcnr 6854 <R cltr 6860 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-br 3846 df-opab 3900 df-suc 4198 df-iom 4406 df-xp 4444 df-1o 6181 df-ni 6861 df-ltr 7274 |
This theorem is referenced by: caucvgsrlemoffval 7339 caucvgsrlemofff 7340 caucvgsrlemoffcau 7341 caucvgsrlemoffgt1 7342 caucvgsrlemoffres 7343 |
Copyright terms: Public domain | W3C validator |