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Mirrors > Home > MPE Home > Th. List > infeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
infeq2 | ⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supeq2 8237 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐶, ◡𝑅)) | |
2 | df-inf 8232 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
3 | df-inf 8232 | . 2 ⊢ inf(𝐴, 𝐶, 𝑅) = sup(𝐴, 𝐶, ◡𝑅) | |
4 | 1, 2, 3 | 3eqtr4g 2669 | 1 ⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ◡ccnv 5037 supcsup 8229 infcinf 8230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-uni 4373 df-sup 8231 df-inf 8232 |
This theorem is referenced by: (None) |
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