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Mirrors > Home > ILE Home > Th. List > nnmass | Unicode version |
Description: Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnmass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5844 | . . . . . 6 | |
2 | oveq2 5844 | . . . . . . 7 | |
3 | 2 | oveq2d 5852 | . . . . . 6 |
4 | 1, 3 | eqeq12d 2179 | . . . . 5 |
5 | 4 | imbi2d 229 | . . . 4 |
6 | oveq2 5844 | . . . . . 6 | |
7 | oveq2 5844 | . . . . . . 7 | |
8 | 7 | oveq2d 5852 | . . . . . 6 |
9 | 6, 8 | eqeq12d 2179 | . . . . 5 |
10 | oveq2 5844 | . . . . . 6 | |
11 | oveq2 5844 | . . . . . . 7 | |
12 | 11 | oveq2d 5852 | . . . . . 6 |
13 | 10, 12 | eqeq12d 2179 | . . . . 5 |
14 | oveq2 5844 | . . . . . 6 | |
15 | oveq2 5844 | . . . . . . 7 | |
16 | 15 | oveq2d 5852 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2179 | . . . . 5 |
18 | nnmcl 6440 | . . . . . . 7 | |
19 | nnm0 6434 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | nnm0 6434 | . . . . . . . 8 | |
22 | 21 | oveq2d 5852 | . . . . . . 7 |
23 | nnm0 6434 | . . . . . . 7 | |
24 | 22, 23 | sylan9eqr 2219 | . . . . . 6 |
25 | 20, 24 | eqtr4d 2200 | . . . . 5 |
26 | oveq1 5843 | . . . . . . . . 9 | |
27 | nnmsuc 6436 | . . . . . . . . . . . 12 | |
28 | 18, 27 | sylan 281 | . . . . . . . . . . 11 |
29 | 28 | 3impa 1183 | . . . . . . . . . 10 |
30 | nnmsuc 6436 | . . . . . . . . . . . . 13 | |
31 | 30 | 3adant1 1004 | . . . . . . . . . . . 12 |
32 | 31 | oveq2d 5852 | . . . . . . . . . . 11 |
33 | nnmcl 6440 | . . . . . . . . . . . . . . . . 17 | |
34 | nndi 6445 | . . . . . . . . . . . . . . . . 17 | |
35 | 33, 34 | syl3an2 1261 | . . . . . . . . . . . . . . . 16 |
36 | 35 | 3exp 1191 | . . . . . . . . . . . . . . 15 |
37 | 36 | expd 256 | . . . . . . . . . . . . . 14 |
38 | 37 | com34 83 | . . . . . . . . . . . . 13 |
39 | 38 | pm2.43d 50 | . . . . . . . . . . . 12 |
40 | 39 | 3imp 1182 | . . . . . . . . . . 11 |
41 | 32, 40 | eqtrd 2197 | . . . . . . . . . 10 |
42 | 29, 41 | eqeq12d 2179 | . . . . . . . . 9 |
43 | 26, 42 | syl5ibr 155 | . . . . . . . 8 |
44 | 43 | 3exp 1191 | . . . . . . 7 |
45 | 44 | com3r 79 | . . . . . 6 |
46 | 45 | impd 252 | . . . . 5 |
47 | 9, 13, 17, 25, 46 | finds2 4572 | . . . 4 |
48 | 5, 47 | vtoclga 2787 | . . 3 |
49 | 48 | expdcom 1429 | . 2 |
50 | 49 | 3imp 1182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 c0 3404 csuc 4337 com 4561 (class class class)co 5836 coa 6372 comu 6373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-oadd 6379 df-omul 6380 |
This theorem is referenced by: mulasspig 7264 enq0tr 7366 addcmpblnq0 7375 mulcmpblnq0 7376 mulcanenq0ec 7377 distrnq0 7391 addassnq0 7394 |
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