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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 8912 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐶, ◡𝑅)) | |
2 | df-inf 8907 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
3 | df-inf 8907 | . 2 ⊢ inf(𝐴, 𝐶, 𝑅) = sup(𝐴, 𝐶, ◡𝑅) | |
4 | 1, 2, 3 | 3eqtr4g 2881 | 1 ⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ◡ccnv 5554 supcsup 8904 infcinf 8905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-uni 4839 df-sup 8906 df-inf 8907 |
This theorem is referenced by: (None) |
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