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Mirrors > Home > ILE Home > Th. List > caucvgsrlemasr | GIF version |
Description: Lemma for caucvgsr 7634. 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 7570 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
3 | 2 | brel 4599 | . . . . 5 ⊢ (𝐴 <R (𝐹‘𝑚) → (𝐴 ∈ R ∧ (𝐹‘𝑚) ∈ R)) |
4 | 3 | simpld 111 | . . . 4 ⊢ (𝐴 <R (𝐹‘𝑚) → 𝐴 ∈ R) |
5 | 4 | ralimi 2498 | . . 3 ⊢ (∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚) → ∀𝑚 ∈ N 𝐴 ∈ R) |
6 | 1, 5 | syl 14 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 ∈ R) |
7 | 1pi 7147 | . . 3 ⊢ 1o ∈ N | |
8 | elex2 2705 | . . 3 ⊢ (1o ∈ N → ∃𝑥 𝑥 ∈ N) | |
9 | r19.3rmv 3458 | . . 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 1469 ∈ wcel 1481 ∀wral 2417 class class class wbr 3937 ‘cfv 5131 1oc1o 6314 Ncnpi 7104 Rcnr 7129 <R cltr 7135 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-suc 4301 df-iom 4513 df-xp 4553 df-1o 6321 df-ni 7136 df-ltr 7562 |
This theorem is referenced by: caucvgsrlemoffval 7628 caucvgsrlemofff 7629 caucvgsrlemoffcau 7630 caucvgsrlemoffgt1 7631 caucvgsrlemoffres 7632 |
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