![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > assintopmap | Structured version Visualization version GIF version |
Description: The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
assintopmap | ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | assintopval 42635 | . 2 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}) | |
2 | clintopval 42634 | . . 3 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀))) | |
3 | rabeq 3377 | . . 3 ⊢ (( clIntOp ‘𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀)) → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝑀 ∈ 𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}) |
5 | 1, 4 | eqtrd 2834 | 1 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 {crab 3094 class class class wbr 4844 × cxp 5311 ‘cfv 6102 (class class class)co 6879 ↑𝑚 cmap 8096 assLaw casslaw 42614 clIntOp cclintop 42627 assIntOp cassintop 42628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-iota 6065 df-fun 6104 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-intop 42629 df-clintop 42630 df-assintop 42631 |
This theorem is referenced by: assintop 42639 isassintop 42640 |
Copyright terms: Public domain | W3C validator |