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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 9485 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, ◡𝑅) = sup(𝐶, 𝐴, ◡𝑅)) | |
| 2 | df-inf 9483 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
| 3 | df-inf 9483 | . 2 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅) | |
| 4 | 1, 2, 3 | 3eqtr4g 2802 | 1 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ◡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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-ss 3968 df-uni 4908 df-sup 9482 df-inf 9483 |
| This theorem is referenced by: infeq1d 9517 infeq1i 9518 ramcl2lem 17047 odfval 19550 odval 19552 submod 19587 ioorval 25609 uniioombllem6 25623 infleinf 45383 infxrpnf 45457 prproropf1olem2 47491 prproropf1olem3 47492 prproropf1olem4 47493 prproropf1o 47494 prproropreud 47496 |
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