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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 5876 | . . 3 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
2 | supeq3 9473 | . . 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 9467 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
5 | df-inf 9467 | . 2 ⊢ inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, ◡𝑆) | |
6 | 3, 4, 5 | 3eqtr4g 2793 | 1 ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ◡ccnv 5677 supcsup 9464 infcinf 9465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-in 3954 df-ss 3964 df-uni 4909 df-br 5149 df-opab 5211 df-cnv 5686 df-sup 9466 df-inf 9467 |
This theorem is referenced by: (None) |
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