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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 5741 | . . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹) |
5 | 1, 2, 4 | supeq123d 8908 | . 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹)) |
6 | df-inf 8901 | . 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶) | |
7 | df-inf 8901 | . 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹) | |
8 | 5, 6, 7 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ◡ccnv 5549 supcsup 8898 infcinf 8899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-in 3943 df-ss 3952 df-uni 4833 df-br 5060 df-opab 5122 df-cnv 5558 df-sup 8900 df-inf 8901 |
This theorem is referenced by: wsuceq123 33096 wlimeq12 33101 |
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