Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > addcmpblnq | Unicode version |
Description: Lemma showing compatibility of addition. (Contributed by NM, 27-Aug-1995.) |
Ref | Expression |
---|---|
addcmpblnq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distrpig 7295 | . . . . . . . 8 | |
2 | 1 | adantl 275 | . . . . . . 7 |
3 | simplll 528 | . . . . . . . 8 | |
4 | simprlr 533 | . . . . . . . 8 | |
5 | mulclpi 7290 | . . . . . . . 8 | |
6 | 3, 4, 5 | syl2anc 409 | . . . . . . 7 |
7 | simpllr 529 | . . . . . . . 8 | |
8 | simprll 532 | . . . . . . . 8 | |
9 | mulclpi 7290 | . . . . . . . 8 | |
10 | 7, 8, 9 | syl2anc 409 | . . . . . . 7 |
11 | mulclpi 7290 | . . . . . . . . 9 | |
12 | 11 | ad2ant2l 505 | . . . . . . . 8 |
13 | 12 | ad2ant2l 505 | . . . . . . 7 |
14 | addclpi 7289 | . . . . . . . 8 | |
15 | 14 | adantl 275 | . . . . . . 7 |
16 | mulcompig 7293 | . . . . . . . 8 | |
17 | 16 | adantl 275 | . . . . . . 7 |
18 | 2, 6, 10, 13, 15, 17 | caovdir2d 6029 | . . . . . 6 |
19 | simplrr 531 | . . . . . . . 8 | |
20 | mulasspig 7294 | . . . . . . . . 9 | |
21 | 20 | adantl 275 | . . . . . . . 8 |
22 | simprrr 535 | . . . . . . . 8 | |
23 | mulclpi 7290 | . . . . . . . . 9 | |
24 | 23 | adantl 275 | . . . . . . . 8 |
25 | 3, 4, 19, 17, 21, 22, 24 | caov4d 6037 | . . . . . . 7 |
26 | 7, 8, 19, 17, 21, 22, 24 | caov4d 6037 | . . . . . . 7 |
27 | 25, 26 | oveq12d 5871 | . . . . . 6 |
28 | 18, 27 | eqtrd 2203 | . . . . 5 |
29 | oveq1 5860 | . . . . . 6 | |
30 | oveq2 5861 | . . . . . 6 | |
31 | 29, 30 | oveqan12d 5872 | . . . . 5 |
32 | 28, 31 | sylan9eq 2223 | . . . 4 |
33 | mulclpi 7290 | . . . . . . . 8 | |
34 | 7, 4, 33 | syl2anc 409 | . . . . . . 7 |
35 | simplrl 530 | . . . . . . . 8 | |
36 | mulclpi 7290 | . . . . . . . 8 | |
37 | 35, 22, 36 | syl2anc 409 | . . . . . . 7 |
38 | simprrl 534 | . . . . . . . 8 | |
39 | mulclpi 7290 | . . . . . . . 8 | |
40 | 19, 38, 39 | syl2anc 409 | . . . . . . 7 |
41 | distrpig 7295 | . . . . . . 7 | |
42 | 34, 37, 40, 41 | syl3anc 1233 | . . . . . 6 |
43 | 7, 4, 35, 17, 21, 22, 24 | caov4d 6037 | . . . . . . 7 |
44 | 7, 4, 19, 17, 21, 38, 24 | caov4d 6037 | . . . . . . 7 |
45 | 43, 44 | oveq12d 5871 | . . . . . 6 |
46 | 42, 45 | eqtrd 2203 | . . . . 5 |
47 | 46 | adantr 274 | . . . 4 |
48 | 32, 47 | eqtr4d 2206 | . . 3 |
49 | addclpi 7289 | . . . . . . . . . 10 | |
50 | 5, 9, 49 | syl2an 287 | . . . . . . . . 9 |
51 | 50 | an42s 584 | . . . . . . . 8 |
52 | 33 | ad2ant2l 505 | . . . . . . . 8 |
53 | 51, 52 | jca 304 | . . . . . . 7 |
54 | addclpi 7289 | . . . . . . . . . 10 | |
55 | 36, 39, 54 | syl2an 287 | . . . . . . . . 9 |
56 | 55 | an42s 584 | . . . . . . . 8 |
57 | 56, 12 | jca 304 | . . . . . . 7 |
58 | 53, 57 | anim12i 336 | . . . . . 6 |
59 | 58 | an4s 583 | . . . . 5 |
60 | enqbreq 7318 | . . . . 5 | |
61 | 59, 60 | syl 14 | . . . 4 |
62 | 61 | adantr 274 | . . 3 |
63 | 48, 62 | mpbird 166 | . 2 |
64 | 63 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 cop 3586 class class class wbr 3989 (class class class)co 5853 cnpi 7234 cpli 7235 cmi 7236 ceq 7241 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-omul 6400 df-ni 7266 df-pli 7267 df-mi 7268 df-enq 7309 |
This theorem is referenced by: addpipqqs 7332 |
Copyright terms: Public domain | W3C validator |