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Mirrors > Home > MPE Home > Th. List > Mathboxes > clcllaw | Structured version Visualization version GIF version |
Description: Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.) |
Ref | Expression |
---|---|
clcllaw | ⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cllaw 44113 | . . . 4 ⊢ clLaw = {〈𝑜, 𝑚〉 ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚} | |
2 | 1 | bropaex12 5642 | . . 3 ⊢ ( ⚬ clLaw 𝑀 → ( ⚬ ∈ V ∧ 𝑀 ∈ V)) |
3 | iscllaw 44116 | . . . 4 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀)) | |
4 | ovrspc2v 7182 | . . . . 5 ⊢ (((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀) | |
5 | 4 | expcom 416 | . . . 4 ⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)) |
6 | 3, 5 | syl6bi 255 | . . 3 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀))) |
7 | 2, 6 | mpcom 38 | . 2 ⊢ ( ⚬ clLaw 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)) |
8 | 7 | 3impib 1112 | 1 ⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 class class class wbr 5066 (class class class)co 7156 clLaw ccllaw 44110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-iota 6314 df-fv 6363 df-ov 7159 df-cllaw 44113 |
This theorem is referenced by: (None) |
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