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Mirrors > Home > ILE Home > Th. List > genipv | Unicode version |
Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.) |
Ref | Expression |
---|---|
genp.1 | |
genp.2 |
Ref | Expression |
---|---|
genipv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5781 | . . . 4 | |
2 | fveq2 5421 | . . . . . . 7 | |
3 | 2 | rexeqdv 2633 | . . . . . 6 |
4 | 3 | rabbidv 2675 | . . . . 5 |
5 | fveq2 5421 | . . . . . . 7 | |
6 | 5 | rexeqdv 2633 | . . . . . 6 |
7 | 6 | rabbidv 2675 | . . . . 5 |
8 | 4, 7 | opeq12d 3713 | . . . 4 |
9 | 1, 8 | eqeq12d 2154 | . . 3 |
10 | oveq2 5782 | . . . 4 | |
11 | fveq2 5421 | . . . . . . . 8 | |
12 | 11 | rexeqdv 2633 | . . . . . . 7 |
13 | 12 | rexbidv 2438 | . . . . . 6 |
14 | 13 | rabbidv 2675 | . . . . 5 |
15 | fveq2 5421 | . . . . . . . 8 | |
16 | 15 | rexeqdv 2633 | . . . . . . 7 |
17 | 16 | rexbidv 2438 | . . . . . 6 |
18 | 17 | rabbidv 2675 | . . . . 5 |
19 | 14, 18 | opeq12d 3713 | . . . 4 |
20 | 10, 19 | eqeq12d 2154 | . . 3 |
21 | nqex 7171 | . . . . . . 7 | |
22 | 21 | a1i 9 | . . . . . 6 |
23 | rabssab 3184 | . . . . . . 7 | |
24 | prop 7283 | . . . . . . . . . . . 12 | |
25 | elprnql 7289 | . . . . . . . . . . . 12 | |
26 | 24, 25 | sylan 281 | . . . . . . . . . . 11 |
27 | prop 7283 | . . . . . . . . . . . 12 | |
28 | elprnql 7289 | . . . . . . . . . . . 12 | |
29 | 27, 28 | sylan 281 | . . . . . . . . . . 11 |
30 | genp.2 | . . . . . . . . . . . 12 | |
31 | eleq1 2202 | . . . . . . . . . . . 12 | |
32 | 30, 31 | syl5ibrcom 156 | . . . . . . . . . . 11 |
33 | 26, 29, 32 | syl2an 287 | . . . . . . . . . 10 |
34 | 33 | an4s 577 | . . . . . . . . 9 |
35 | 34 | rexlimdvva 2557 | . . . . . . . 8 |
36 | 35 | abssdv 3171 | . . . . . . 7 |
37 | 23, 36 | sstrid 3108 | . . . . . 6 |
38 | 22, 37 | ssexd 4068 | . . . . 5 |
39 | rabssab 3184 | . . . . . . 7 | |
40 | elprnqu 7290 | . . . . . . . . . . . 12 | |
41 | 24, 40 | sylan 281 | . . . . . . . . . . 11 |
42 | elprnqu 7290 | . . . . . . . . . . . 12 | |
43 | 27, 42 | sylan 281 | . . . . . . . . . . 11 |
44 | 41, 43, 32 | syl2an 287 | . . . . . . . . . 10 |
45 | 44 | an4s 577 | . . . . . . . . 9 |
46 | 45 | rexlimdvva 2557 | . . . . . . . 8 |
47 | 46 | abssdv 3171 | . . . . . . 7 |
48 | 39, 47 | sstrid 3108 | . . . . . 6 |
49 | 22, 48 | ssexd 4068 | . . . . 5 |
50 | opelxp 4569 | . . . . 5 | |
51 | 38, 49, 50 | sylanbrc 413 | . . . 4 |
52 | fveq2 5421 | . . . . . . . 8 | |
53 | 52 | rexeqdv 2633 | . . . . . . 7 |
54 | 53 | rabbidv 2675 | . . . . . 6 |
55 | fveq2 5421 | . . . . . . . 8 | |
56 | 55 | rexeqdv 2633 | . . . . . . 7 |
57 | 56 | rabbidv 2675 | . . . . . 6 |
58 | 54, 57 | opeq12d 3713 | . . . . 5 |
59 | fveq2 5421 | . . . . . . . . 9 | |
60 | 59 | rexeqdv 2633 | . . . . . . . 8 |
61 | 60 | rexbidv 2438 | . . . . . . 7 |
62 | 61 | rabbidv 2675 | . . . . . 6 |
63 | fveq2 5421 | . . . . . . . . 9 | |
64 | 63 | rexeqdv 2633 | . . . . . . . 8 |
65 | 64 | rexbidv 2438 | . . . . . . 7 |
66 | 65 | rabbidv 2675 | . . . . . 6 |
67 | 62, 66 | opeq12d 3713 | . . . . 5 |
68 | genp.1 | . . . . . 6 | |
69 | 68 | genpdf 7316 | . . . . 5 |
70 | 58, 67, 69 | ovmpog 5905 | . . . 4 |
71 | 51, 70 | mpd3an3 1316 | . . 3 |
72 | 9, 20, 71 | vtocl2ga 2754 | . 2 |
73 | eqeq1 2146 | . . . . . 6 | |
74 | 73 | 2rexbidv 2460 | . . . . 5 |
75 | oveq1 5781 | . . . . . . 7 | |
76 | 75 | eqeq2d 2151 | . . . . . 6 |
77 | oveq2 5782 | . . . . . . 7 | |
78 | 77 | eqeq2d 2151 | . . . . . 6 |
79 | 76, 78 | cbvrex2v 2666 | . . . . 5 |
80 | 74, 79 | syl6bb 195 | . . . 4 |
81 | 80 | cbvrabv 2685 | . . 3 |
82 | 73 | 2rexbidv 2460 | . . . . 5 |
83 | 76, 78 | cbvrex2v 2666 | . . . . 5 |
84 | 82, 83 | syl6bb 195 | . . . 4 |
85 | 84 | cbvrabv 2685 | . . 3 |
86 | 81, 85 | opeq12i 3710 | . 2 |
87 | 72, 86 | syl6eq 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 cab 2125 wrex 2417 crab 2420 cvv 2686 cop 3530 cxp 4537 cfv 5123 (class class class)co 5774 cmpo 5776 c1st 6036 c2nd 6037 cnq 7088 cnp 7099 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-qs 6435 df-ni 7112 df-nqqs 7156 df-inp 7274 |
This theorem is referenced by: genpelvl 7320 genpelvu 7321 plpvlu 7346 mpvlu 7347 |
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