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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clintopcllaw | Structured version Visualization version GIF version |
Description: The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
clintopcllaw | ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clintop 47378 | . . 3 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀) | |
2 | ffnov 7541 | . . . 4 ⊢ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ↔ ( ⚬ Fn (𝑀 × 𝑀) ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀)) | |
3 | 2 | simprbi 495 | . . 3 ⊢ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀) |
4 | 1, 3 | syl 17 | . 2 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀) |
5 | elfvex 6928 | . . 3 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V) | |
6 | iscllaw 47359 | . . 3 ⊢ (( ⚬ ∈ ( clIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀)) | |
7 | 5, 6 | mpdan 685 | . 2 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀)) |
8 | 4, 7 | mpbird 256 | 1 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 class class class wbr 5144 × cxp 5671 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 clLaw ccllaw 47353 clIntOp cclintop 47367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 df-map 8840 df-cllaw 47356 df-intop 47369 df-clintop 47370 |
This theorem is referenced by: assintopcllaw 47382 |
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