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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atlem9 | Structured version Visualization version Unicode version |
Description: Lemma for 4at 34899. Substitute for . (Contributed by NM, 9-Jul-2012.) |
Ref | Expression |
---|---|
4at.l | |
4at.j | |
4at.a |
Ref | Expression |
---|---|
4atlem9 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp11 1091 | . . 3 | |
2 | simp22 1095 | . . 3 | |
3 | simp23 1096 | . . 3 | |
4 | hllat 34650 | . . . . 5 | |
5 | 1, 4 | syl 17 | . . . 4 |
6 | simp1 1061 | . . . . 5 | |
7 | eqid 2622 | . . . . . 6 | |
8 | 4at.j | . . . . . 6 | |
9 | 4at.a | . . . . . 6 | |
10 | 7, 8, 9 | hlatjcl 34653 | . . . . 5 |
11 | 6, 10 | syl 17 | . . . 4 |
12 | simp21 1094 | . . . . 5 | |
13 | 7, 9 | atbase 34576 | . . . . 5 |
14 | 12, 13 | syl 17 | . . . 4 |
15 | 7, 8 | latjcl 17051 | . . . 4 |
16 | 5, 11, 14, 15 | syl3anc 1326 | . . 3 |
17 | simp3 1063 | . . 3 | |
18 | 4at.l | . . . 4 | |
19 | 7, 18, 8, 9 | hlexchb2 34671 | . . 3 |
20 | 1, 2, 3, 16, 17, 19 | syl131anc 1339 | . 2 |
21 | 18, 8, 9 | 4atlem4d 34888 | . . . 4 |
22 | 6, 12, 3, 21 | syl12anc 1324 | . . 3 |
23 | 22 | breq2d 4665 | . 2 |
24 | 18, 8, 9 | 4atlem4d 34888 | . . . 4 |
25 | 6, 12, 2, 24 | syl12anc 1324 | . . 3 |
26 | 25, 22 | eqeq12d 2637 | . 2 |
27 | 20, 23, 26 | 3bitr4d 300 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 w3a 1037 wceq 1483 wcel 1990 class class class wbr 4653 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: 4atlem10b 34891 |
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