| 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 9483 | . 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 5889 | . . 3 ⊢ Ⅎ𝑥◡𝑅 |
| 6 | 2, 3, 5 | nfsup 9491 | . 2 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, ◡𝑅) |
| 7 | 1, 6 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: Ⅎwnfc 2890 ◡ccnv 5684 supcsup 9480 infcinf 9481 |
| 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-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-sup 9482 df-inf 9483 |
| This theorem is referenced by: iundisj 25583 iundisjf 32602 iundisjfi 32798 nfwsuc 35819 nfwlim 35823 allbutfiinf 45431 infrpgernmpt 45476 liminflelimsuplem 45790 stoweidlem62 46077 fourierdlem31 46153 iunhoiioolem 46690 smfinf 46833 prmdvdsfmtnof1lem1 47571 |
| Copyright terms: Public domain | W3C validator |