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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atlem4a | Structured version Visualization version Unicode version |
Description: Lemma for 4at 34899. Frequently used associative law. (Contributed by NM, 9-Jul-2012.) |
Ref | Expression |
---|---|
4at.l | |
4at.j | |
4at.a |
Ref | Expression |
---|---|
4atlem4a |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1064 | . . . 4 | |
2 | hllat 34650 | . . . 4 | |
3 | 1, 2 | syl 17 | . . 3 |
4 | simpl2 1065 | . . . 4 | |
5 | eqid 2622 | . . . . 5 | |
6 | 4at.a | . . . . 5 | |
7 | 5, 6 | atbase 34576 | . . . 4 |
8 | 4, 7 | syl 17 | . . 3 |
9 | simpl3 1066 | . . . 4 | |
10 | 5, 6 | atbase 34576 | . . . 4 |
11 | 9, 10 | syl 17 | . . 3 |
12 | simprl 794 | . . . 4 | |
13 | simprr 796 | . . . 4 | |
14 | 4at.j | . . . . 5 | |
15 | 5, 14, 6 | hlatjcl 34653 | . . . 4 |
16 | 1, 12, 13, 15 | syl3anc 1326 | . . 3 |
17 | 5, 14 | latjass 17095 | . . 3 |
18 | 3, 8, 11, 16, 17 | syl13anc 1328 | . 2 |
19 | 14, 6 | hlatjass 34656 | . . . 4 |
20 | 1, 9, 12, 13, 19 | syl13anc 1328 | . . 3 |
21 | 20 | oveq2d 6666 | . 2 |
22 | 18, 21 | eqtr4d 2659 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cple 15948 cjn 16944 clat 17045 catm 34550 chlt 34637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-preset 16928 df-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 |
This theorem is referenced by: 4atlem12a 34896 |
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