Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > pmodN | Structured version Visualization version Unicode version |
Description: The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pmod.a | |
pmod.s | |
pmod.p |
Ref | Expression |
---|---|
pmodN |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3805 | . 2 | |
2 | hllat 34650 | . . . . 5 | |
3 | 2 | adantr 481 | . . . 4 |
4 | simpr2 1068 | . . . 4 | |
5 | inss2 3834 | . . . . 5 | |
6 | simpr3 1069 | . . . . 5 | |
7 | 5, 6 | syl5ss 3614 | . . . 4 |
8 | pmod.a | . . . . 5 | |
9 | pmod.p | . . . . 5 | |
10 | 8, 9 | paddcom 35099 | . . . 4 |
11 | 3, 4, 7, 10 | syl3anc 1326 | . . 3 |
12 | 11 | ineq2d 3814 | . 2 |
13 | incom 3805 | . . . 4 | |
14 | 13 | oveq2i 6661 | . . 3 |
15 | inss2 3834 | . . . . 5 | |
16 | 15, 4 | syl5ss 3614 | . . . 4 |
17 | 8, 9 | paddcom 35099 | . . . 4 |
18 | 3, 16, 7, 17 | syl3anc 1326 | . . 3 |
19 | simpr1 1067 | . . . . 5 | |
20 | 7, 4, 19 | 3jca 1242 | . . . 4 |
21 | inss1 3833 | . . . . 5 | |
22 | pmod.s | . . . . . 6 | |
23 | 8, 22, 9 | pmod1i 35134 | . . . . 5 |
24 | 21, 23 | mpi 20 | . . . 4 |
25 | 20, 24 | syldan 487 | . . 3 |
26 | 14, 18, 25 | 3eqtr4a 2682 | . 2 |
27 | 1, 12, 26 | 3eqtr4a 2682 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cin 3573 wss 3574 cfv 5888 (class class class)co 6650 clat 17045 catm 34550 chlt 34637 cpsubsp 34782 cpadd 35081 |
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-mpt2 6655 df-1st 7168 df-2nd 7169 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-psubsp 34789 df-padd 35082 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |