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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 4772 | . . 3 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
2 | supeq3 6946 | . . 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 6941 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
5 | df-inf 6941 | . 2 ⊢ inf(𝐴, 𝐵, 𝑆) = sup(𝐴, 𝐵, ◡𝑆) | |
6 | 3, 4, 5 | 3eqtr4g 2222 | 1 ⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ◡ccnv 4597 supcsup 6938 infcinf 6939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-in 3117 df-ss 3124 df-uni 3784 df-br 3977 df-opab 4038 df-cnv 4606 df-sup 6940 df-inf 6941 |
This theorem is referenced by: (None) |
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