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Mirrors > Home > MPE Home > Th. List > infeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
infeq1 | ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supeq1 9454 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, ◡𝑅) = sup(𝐶, 𝐴, ◡𝑅)) | |
2 | df-inf 9452 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
3 | df-inf 9452 | . 2 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅) | |
4 | 1, 2, 3 | 3eqtr4g 2792 | 1 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ◡ccnv 5671 supcsup 9449 infcinf 9450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-in 3951 df-ss 3961 df-uni 4904 df-sup 9451 df-inf 9452 |
This theorem is referenced by: infeq1d 9486 infeq1i 9487 ramcl2lem 16963 odfval 19471 odval 19473 submod 19508 ioorval 25477 uniioombllem6 25491 infleinf 44667 infxrpnf 44741 prproropf1olem2 46757 prproropf1olem3 46758 prproropf1olem4 46759 prproropf1o 46760 prproropreud 46762 |
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