Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > infeq3 | Structured version Visualization version GIF version |
Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
infeq3 | ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5737 | . . 3 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
2 | supeq3 8901 | . . 3 ⊢ (◡𝑅 = ◡𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆)) |
4 | df-inf 8895 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
5 | df-inf 8895 | . 2 ⊢ inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, ◡𝑆) | |
6 | 3, 4, 5 | 3eqtr4g 2878 | 1 ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ◡ccnv 5547 supcsup 8892 infcinf 8893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-in 3940 df-ss 3949 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-sup 8894 df-inf 8895 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |