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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atlem4d | Unicode version |
Description: Lemma for 4at 30099. Frequently used associative law. (Contributed by NM, 9-Jul-2012.) |
Ref | Expression |
---|---|
4at.l |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4at.j |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4at.a |
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Ref | Expression |
---|---|
4atlem4d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 960 |
. . . 4
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2 | hllat 29850 |
. . . 4
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3 | 1, 2 | syl 16 |
. . 3
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4 | eqid 2408 |
. . . . 5
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5 | 4at.j |
. . . . 5
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6 | 4at.a |
. . . . 5
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7 | 4, 5, 6 | hlatjcl 29853 |
. . . 4
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8 | 7 | adantr 452 |
. . 3
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9 | 4, 6 | atbase 29776 |
. . . 4
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10 | 9 | ad2antrl 709 |
. . 3
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11 | 4, 6 | atbase 29776 |
. . . 4
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12 | 11 | ad2antll 710 |
. . 3
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13 | 4, 5 | latjass 14483 |
. . 3
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14 | 3, 8, 10, 12, 13 | syl13anc 1186 |
. 2
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15 | 4, 5 | latjcl 14438 |
. . . 4
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16 | 3, 8, 10, 15 | syl3anc 1184 |
. . 3
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17 | 4, 5 | latjcom 14447 |
. . 3
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18 | 3, 16, 12, 17 | syl3anc 1184 |
. 2
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19 | 14, 18 | eqtr3d 2442 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: 4atlem9 30089 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-rep 4284 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-nel 2574 df-ral 2675 df-rex 2676 df-reu 2677 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-op 3787 df-uni 3980 df-iun 4059 df-br 4177 df-opab 4231 df-mpt 4232 df-id 4462 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-oprab 6048 df-mpt2 6049 df-1st 6312 df-2nd 6313 df-undef 6506 df-riota 6512 df-poset 14362 df-lub 14390 df-join 14392 df-lat 14434 df-ats 29754 df-atl 29785 df-cvlat 29809 df-hlat 29838 |
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