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Mirrors > Home > ILE Home > Th. List > mullocprlem | Unicode version |
Description: Calculations for mullocpr 7503. (Contributed by Jim Kingdon, 10-Dec-2019.) |
Ref | Expression |
---|---|
mullocprlem.ab | |
mullocprlem.uqedu | |
mullocprlem.edutdu | |
mullocprlem.tdudr | |
mullocprlem.qr | |
mullocprlem.duq | |
mullocprlem.du | |
mullocprlem.et |
Ref | Expression |
---|---|
mullocprlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mullocprlem.uqedu | . . . . . . 7 | |
2 | mullocprlem.et | . . . . . . . . 9 | |
3 | 2 | simpld 111 | . . . . . . . 8 |
4 | mullocprlem.duq | . . . . . . . . 9 | |
5 | 4 | simpld 111 | . . . . . . . 8 |
6 | 4 | simprd 113 | . . . . . . . 8 |
7 | mulcomnqg 7315 | . . . . . . . . 9 | |
8 | 7 | adantl 275 | . . . . . . . 8 |
9 | mulassnqg 7316 | . . . . . . . . 9 | |
10 | 9 | adantl 275 | . . . . . . . 8 |
11 | 3, 5, 6, 8, 10 | caov13d 6016 | . . . . . . 7 |
12 | 1, 11 | breqtrd 4002 | . . . . . 6 |
13 | mullocprlem.qr | . . . . . . . 8 | |
14 | 13 | simpld 111 | . . . . . . 7 |
15 | mulclnq 7308 | . . . . . . . 8 | |
16 | 5, 3, 15 | syl2anc 409 | . . . . . . 7 |
17 | ltmnqg 7333 | . . . . . . 7 | |
18 | 14, 16, 6, 17 | syl3anc 1227 | . . . . . 6 |
19 | 12, 18 | mpbird 166 | . . . . 5 |
20 | 19 | adantr 274 | . . . 4 |
21 | mullocprlem.ab | . . . . . . . 8 | |
22 | 21 | simpld 111 | . . . . . . 7 |
23 | mullocprlem.du | . . . . . . . 8 | |
24 | 23 | simpld 111 | . . . . . . 7 |
25 | 22, 24 | jca 304 | . . . . . 6 |
26 | 25 | adantr 274 | . . . . 5 |
27 | 21 | simprd 113 | . . . . . 6 |
28 | 27 | anim1i 338 | . . . . 5 |
29 | 14 | adantr 274 | . . . . 5 |
30 | mulnqprl 7500 | . . . . 5 | |
31 | 26, 28, 29, 30 | syl21anc 1226 | . . . 4 |
32 | 20, 31 | mpd 13 | . . 3 |
33 | 32 | orcd 723 | . 2 |
34 | 2 | simprd 113 | . . . . . . 7 |
35 | mulcomnqg 7315 | . . . . . . 7 | |
36 | 34, 6, 35 | syl2anc 409 | . . . . . 6 |
37 | mullocprlem.tdudr | . . . . . . 7 | |
38 | mulclnq 7308 | . . . . . . . . . 10 | |
39 | 34, 6, 38 | syl2anc 409 | . . . . . . . . 9 |
40 | 13 | simprd 113 | . . . . . . . . 9 |
41 | ltmnqg 7333 | . . . . . . . . 9 | |
42 | 39, 40, 5, 41 | syl3anc 1227 | . . . . . . . 8 |
43 | 34, 5, 6, 8, 10 | caov12d 6014 | . . . . . . . . 9 |
44 | 43 | breq1d 3986 | . . . . . . . 8 |
45 | 42, 44 | bitr4d 190 | . . . . . . 7 |
46 | 37, 45 | mpbird 166 | . . . . . 6 |
47 | 36, 46 | eqbrtrrd 4000 | . . . . 5 |
48 | 47 | adantr 274 | . . . 4 |
49 | 23 | simprd 113 | . . . . . . 7 |
50 | 22, 49 | jca 304 | . . . . . 6 |
51 | 50 | adantr 274 | . . . . 5 |
52 | 27 | anim1i 338 | . . . . 5 |
53 | 40 | adantr 274 | . . . . 5 |
54 | mulnqpru 7501 | . . . . 5 | |
55 | 51, 52, 53, 54 | syl21anc 1226 | . . . 4 |
56 | 48, 55 | mpd 13 | . . 3 |
57 | 56 | olcd 724 | . 2 |
58 | mullocprlem.edutdu | . . . 4 | |
59 | mulclnq 7308 | . . . . . . 7 | |
60 | 4, 59 | syl 14 | . . . . . 6 |
61 | ltmnqg 7333 | . . . . . 6 | |
62 | 3, 34, 60, 61 | syl3anc 1227 | . . . . 5 |
63 | mulcomnqg 7315 | . . . . . . 7 | |
64 | 60, 3, 63 | syl2anc 409 | . . . . . 6 |
65 | mulcomnqg 7315 | . . . . . . 7 | |
66 | 60, 34, 65 | syl2anc 409 | . . . . . 6 |
67 | 64, 66 | breq12d 3989 | . . . . 5 |
68 | 62, 67 | bitrd 187 | . . . 4 |
69 | 58, 68 | mpbird 166 | . . 3 |
70 | prop 7407 | . . . 4 | |
71 | prloc 7423 | . . . 4 | |
72 | 70, 71 | sylan 281 | . . 3 |
73 | 27, 69, 72 | syl2anc 409 | . 2 |
74 | 33, 57, 73 | mpjaodan 788 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 w3a 967 wceq 1342 wcel 2135 cop 3573 class class class wbr 3976 cfv 5182 (class class class)co 5836 c1st 6098 c2nd 6099 cnq 7212 cmq 7215 cltq 7217 cnp 7223 cmp 7226 |
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-dc 825 df-3or 968 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-eprel 4261 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-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-mi 7238 df-lti 7239 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-inp 7398 df-imp 7401 |
This theorem is referenced by: mullocpr 7503 |
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