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Mirrors > Home > MPE Home > Th. List > Mathboxes > assintop | Structured version Visualization version GIF version |
Description: An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
assintop | ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6131 | . 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V) | |
2 | assintopmap 41632 | . . . 4 ⊢ (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}) | |
3 | 2 | eleq2d 2673 | . . 3 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})) |
4 | breq1 4586 | . . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀)) | |
5 | 4 | elrab 3331 | . . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)) |
6 | elmapi 7765 | . . . . 5 ⊢ ( ⚬ ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀) | |
7 | 6 | anim1i 590 | . . . 4 ⊢ (( ⚬ ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀)) |
8 | 5, 7 | sylbi 206 | . . 3 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀)) |
9 | 3, 8 | syl6bi 242 | . 2 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))) |
10 | 1, 9 | mpcom 37 | 1 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 {crab 2900 Vcvv 3173 class class class wbr 4583 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 assLaw casslaw 41610 assIntOp cassintop 41624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-intop 41625 df-clintop 41626 df-assintop 41627 |
This theorem is referenced by: assintopasslaw 41639 |
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