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Mirrors > Home > ILE Home > Th. List > infeq3 | 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 4610 | . . 3 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
2 | supeq3 6683 | . . 3 ⊢ (◡𝑅 = ◡𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, ◡𝑅) = sup(𝐴, 𝐵, ◡𝑆)) |
4 | df-inf 6678 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
5 | df-inf 6678 | . 2 ⊢ inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, ◡𝑆) | |
6 | 3, 4, 5 | 3eqtr4g 2145 | 1 ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ◡ccnv 4437 supcsup 6675 infcinf 6676 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-in 3005 df-ss 3012 df-uni 3654 df-br 3846 df-opab 3900 df-cnv 4446 df-sup 6677 df-inf 6678 |
This theorem is referenced by: (None) |
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