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| Mirrors > Home > MPE Home > Th. List > infeq3 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
| Ref | Expression |
|---|---|
| infeq3 | ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnveq 5884 | . . 3 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
| 2 | supeq3 9489 | . . 3 ⊢ (◡𝑅 = ◡𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆)) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆)) |
| 4 | df-inf 9483 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
| 5 | df-inf 9483 | . 2 ⊢ inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, ◡𝑆) | |
| 6 | 3, 4, 5 | 3eqtr4g 2802 | 1 ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) |
| 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-br 5144 df-opab 5206 df-cnv 5693 df-sup 9482 df-inf 9483 |
| This theorem is referenced by: (None) |
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