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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 8642 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐶, ◡𝑅)) | |
2 | df-inf 8637 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
3 | df-inf 8637 | . 2 ⊢ inf(𝐴, 𝐶, 𝑅) = sup(𝐴, 𝐶, ◡𝑅) | |
4 | 1, 2, 3 | 3eqtr4g 2839 | 1 ⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ◡ccnv 5354 supcsup 8634 infcinf 8635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-uni 4672 df-sup 8636 df-inf 8637 |
This theorem is referenced by: (None) |
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