| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5886 | . . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹) |
| 5 | 1, 2, 4 | supeq123d 9490 | . 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹)) |
| 6 | df-inf 9483 | . 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶) | |
| 7 | df-inf 9483 | . 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹) | |
| 8 | 5, 6, 7 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) |
| 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: wsuceq123 35815 wlimeq12 35820 |
| Copyright terms: Public domain | W3C validator |