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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 9306 | . 2 ⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐶, ◡𝑅)) | |
2 | df-inf 9301 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
3 | df-inf 9301 | . 2 ⊢ inf(𝐴, 𝐶, 𝑅) = sup(𝐴, 𝐶, ◡𝑅) | |
4 | 1, 2, 3 | 3eqtr4g 2801 | 1 ⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ◡ccnv 5620 supcsup 9298 infcinf 9299 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rab 3404 df-v 3443 df-in 3905 df-ss 3915 df-uni 4854 df-sup 9300 df-inf 9301 |
This theorem is referenced by: (None) |
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