![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfinf | Structured version Visualization version GIF version |
Description: Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
nfinf.1 | ⊢ Ⅎ𝑥𝐴 |
nfinf.2 | ⊢ Ⅎ𝑥𝐵 |
nfinf.3 | ⊢ Ⅎ𝑥𝑅 |
Ref | Expression |
---|---|
nfinf | ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 8891 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
2 | nfinf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfinf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | nfinf.3 | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
5 | 4 | nfcnv 5713 | . . 3 ⊢ Ⅎ𝑥◡𝑅 |
6 | 2, 3, 5 | nfsup 8899 | . 2 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, ◡𝑅) |
7 | 1, 6 | nfcxfr 2953 | 1 ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2936 ◡ccnv 5518 supcsup 8888 infcinf 8889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-sup 8890 df-inf 8891 |
This theorem is referenced by: iundisj 24152 iundisjf 30352 iundisjfi 30545 nfwsuc 33218 nfwlim 33222 allbutfiinf 42057 infrpgernmpt 42104 liminflelimsuplem 42417 stoweidlem62 42704 fourierdlem31 42780 iunhoiioolem 43314 smfinf 43449 prmdvdsfmtnof1lem1 44101 |
Copyright terms: Public domain | W3C validator |