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Mirrors > Home > MPE Home > Th. List > infeq123d | Structured version Visualization version GIF version |
Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
infeq123d.a | ⊢ (𝜑 → 𝐴 = 𝐷) |
infeq123d.b | ⊢ (𝜑 → 𝐵 = 𝐸) |
infeq123d.c | ⊢ (𝜑 → 𝐶 = 𝐹) |
Ref | Expression |
---|---|
infeq123d | ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infeq123d.a | . . 3 ⊢ (𝜑 → 𝐴 = 𝐷) | |
2 | infeq123d.b | . . 3 ⊢ (𝜑 → 𝐵 = 𝐸) | |
3 | infeq123d.c | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐹) | |
4 | 3 | cnveqd 5298 | . . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹) |
5 | 1, 2, 4 | supeq123d 8356 | . 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹)) |
6 | df-inf 8349 | . 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶) | |
7 | df-inf 8349 | . 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹) | |
8 | 5, 6, 7 | 3eqtr4g 2681 | 1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ◡ccnv 5113 supcsup 8346 infcinf 8347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-in 3581 df-ss 3588 df-uni 4437 df-br 4654 df-opab 4713 df-cnv 5122 df-sup 8348 df-inf 8349 |
This theorem is referenced by: wsuceq123 31760 wlimeq12 31765 |
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