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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 9482 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, ◡𝑅) = sup(𝐶, 𝐴, ◡𝑅)) | |
2 | df-inf 9480 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
3 | df-inf 9480 | . 2 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅) | |
4 | 1, 2, 3 | 3eqtr4g 2799 | 1 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ◡ccnv 5687 supcsup 9477 infcinf 9478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-ss 3979 df-uni 4912 df-sup 9479 df-inf 9480 |
This theorem is referenced by: infeq1d 9514 infeq1i 9515 ramcl2lem 17042 odfval 19564 odval 19566 submod 19601 ioorval 25622 uniioombllem6 25636 infleinf 45321 infxrpnf 45395 prproropf1olem2 47428 prproropf1olem3 47429 prproropf1olem4 47430 prproropf1o 47431 prproropreud 47433 |
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