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Mirrors > Home > MPE Home > Th. List > Mathboxes > assintopcllaw | Structured version Visualization version GIF version |
Description: The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
assintopcllaw | ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6920 | . 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V) | |
2 | assintopval 47093 | . . . . 5 ⊢ (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}) | |
3 | 2 | eleq2d 2811 | . . . 4 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})) |
4 | breq1 5142 | . . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀)) | |
5 | 4 | elrab 3676 | . . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀)) |
6 | 3, 5 | bitrdi 287 | . . 3 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀))) |
7 | clintopcllaw 47099 | . . . 4 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀) | |
8 | 7 | adantr 480 | . . 3 ⊢ (( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀) → ⚬ clLaw 𝑀) |
9 | 6, 8 | syl6bi 253 | . 2 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀)) |
10 | 1, 9 | mpcom 38 | 1 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 {crab 3424 Vcvv 3466 class class class wbr 5139 ‘cfv 6534 clLaw ccllaw 47071 assLaw casslaw 47072 clIntOp cclintop 47085 assIntOp cassintop 47086 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-map 8819 df-cllaw 47074 df-intop 47087 df-clintop 47088 df-assintop 47089 |
This theorem is referenced by: (None) |
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