Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > homdmcoa | Structured version Visualization version Unicode version |
Description: If and , then and are composable. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homdmcoa.o | compa |
homdmcoa.h | Homa |
homdmcoa.f | |
homdmcoa.g |
Ref | Expression |
---|---|
homdmcoa |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 Nat Nat | |
2 | homdmcoa.h | . . . 4 Homa | |
3 | 1, 2 | homarw 16696 | . . 3 Nat |
4 | homdmcoa.f | . . 3 | |
5 | 3, 4 | sseldi 3601 | . 2 Nat |
6 | 1, 2 | homarw 16696 | . . 3 Nat |
7 | homdmcoa.g | . . 3 | |
8 | 6, 7 | sseldi 3601 | . 2 Nat |
9 | 2 | homacd 16691 | . . . 4 coda |
10 | 4, 9 | syl 17 | . . 3 coda |
11 | 2 | homadm 16690 | . . . 4 |
12 | 7, 11 | syl 17 | . . 3 |
13 | 10, 12 | eqtr4d 2659 | . 2 coda |
14 | homdmcoa.o | . . 3 compa | |
15 | 14, 1 | eldmcoa 16715 | . 2 Nat Nat coda |
16 | 5, 8, 13, 15 | syl3anbrc 1246 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 class class class wbr 4653 cdm 5114 cfv 5888 (class class class)co 6650 cdoma 16670 codaccoda 16671 Natcarw 16672 Homachoma 16673 compaccoa 16704 |
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-ot 4186 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-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-doma 16674 df-coda 16675 df-homa 16676 df-arw 16677 df-coa 16706 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |