![]() |
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 5543 | . . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹) |
5 | 1, 2, 4 | supeq123d 8644 | . 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹)) |
6 | df-inf 8637 | . 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶) | |
7 | df-inf 8637 | . 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹) | |
8 | 5, 6, 7 | 3eqtr4g 2838 | 1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ◡ccnv 5354 supcsup 8634 infcinf 8635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-in 3798 df-ss 3805 df-uni 4672 df-br 4887 df-opab 4949 df-cnv 5363 df-sup 8636 df-inf 8637 |
This theorem is referenced by: wsuceq123 32362 wlimeq12 32367 |
Copyright terms: Public domain | W3C validator |