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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 5749 | . . . 4 | |
2 | fveq2 5389 | . . . . . . 7 | |
3 | 2 | rexeqdv 2610 | . . . . . 6 |
4 | 3 | rabbidv 2649 | . . . . 5 |
5 | fveq2 5389 | . . . . . . 7 | |
6 | 5 | rexeqdv 2610 | . . . . . 6 |
7 | 6 | rabbidv 2649 | . . . . 5 |
8 | 4, 7 | opeq12d 3683 | . . . 4 |
9 | 1, 8 | eqeq12d 2132 | . . 3 |
10 | oveq2 5750 | . . . 4 | |
11 | fveq2 5389 | . . . . . . . 8 | |
12 | 11 | rexeqdv 2610 | . . . . . . 7 |
13 | 12 | rexbidv 2415 | . . . . . 6 |
14 | 13 | rabbidv 2649 | . . . . 5 |
15 | fveq2 5389 | . . . . . . . 8 | |
16 | 15 | rexeqdv 2610 | . . . . . . 7 |
17 | 16 | rexbidv 2415 | . . . . . 6 |
18 | 17 | rabbidv 2649 | . . . . 5 |
19 | 14, 18 | opeq12d 3683 | . . . 4 |
20 | 10, 19 | eqeq12d 2132 | . . 3 |
21 | nqex 7139 | . . . . . . 7 | |
22 | 21 | a1i 9 | . . . . . 6 |
23 | rabssab 3154 | . . . . . . 7 | |
24 | prop 7251 | . . . . . . . . . . . 12 | |
25 | elprnql 7257 | . . . . . . . . . . . 12 | |
26 | 24, 25 | sylan 281 | . . . . . . . . . . 11 |
27 | prop 7251 | . . . . . . . . . . . 12 | |
28 | elprnql 7257 | . . . . . . . . . . . 12 | |
29 | 27, 28 | sylan 281 | . . . . . . . . . . 11 |
30 | genp.2 | . . . . . . . . . . . 12 | |
31 | eleq1 2180 | . . . . . . . . . . . 12 | |
32 | 30, 31 | syl5ibrcom 156 | . . . . . . . . . . 11 |
33 | 26, 29, 32 | syl2an 287 | . . . . . . . . . 10 |
34 | 33 | an4s 562 | . . . . . . . . 9 |
35 | 34 | rexlimdvva 2534 | . . . . . . . 8 |
36 | 35 | abssdv 3141 | . . . . . . 7 |
37 | 23, 36 | sstrid 3078 | . . . . . 6 |
38 | 22, 37 | ssexd 4038 | . . . . 5 |
39 | rabssab 3154 | . . . . . . 7 | |
40 | elprnqu 7258 | . . . . . . . . . . . 12 | |
41 | 24, 40 | sylan 281 | . . . . . . . . . . 11 |
42 | elprnqu 7258 | . . . . . . . . . . . 12 | |
43 | 27, 42 | sylan 281 | . . . . . . . . . . 11 |
44 | 41, 43, 32 | syl2an 287 | . . . . . . . . . 10 |
45 | 44 | an4s 562 | . . . . . . . . 9 |
46 | 45 | rexlimdvva 2534 | . . . . . . . 8 |
47 | 46 | abssdv 3141 | . . . . . . 7 |
48 | 39, 47 | sstrid 3078 | . . . . . 6 |
49 | 22, 48 | ssexd 4038 | . . . . 5 |
50 | opelxp 4539 | . . . . 5 | |
51 | 38, 49, 50 | sylanbrc 413 | . . . 4 |
52 | fveq2 5389 | . . . . . . . 8 | |
53 | 52 | rexeqdv 2610 | . . . . . . 7 |
54 | 53 | rabbidv 2649 | . . . . . 6 |
55 | fveq2 5389 | . . . . . . . 8 | |
56 | 55 | rexeqdv 2610 | . . . . . . 7 |
57 | 56 | rabbidv 2649 | . . . . . 6 |
58 | 54, 57 | opeq12d 3683 | . . . . 5 |
59 | fveq2 5389 | . . . . . . . . 9 | |
60 | 59 | rexeqdv 2610 | . . . . . . . 8 |
61 | 60 | rexbidv 2415 | . . . . . . 7 |
62 | 61 | rabbidv 2649 | . . . . . 6 |
63 | fveq2 5389 | . . . . . . . . 9 | |
64 | 63 | rexeqdv 2610 | . . . . . . . 8 |
65 | 64 | rexbidv 2415 | . . . . . . 7 |
66 | 65 | rabbidv 2649 | . . . . . 6 |
67 | 62, 66 | opeq12d 3683 | . . . . 5 |
68 | genp.1 | . . . . . 6 | |
69 | 68 | genpdf 7284 | . . . . 5 |
70 | 58, 67, 69 | ovmpog 5873 | . . . 4 |
71 | 51, 70 | mpd3an3 1301 | . . 3 |
72 | 9, 20, 71 | vtocl2ga 2728 | . 2 |
73 | eqeq1 2124 | . . . . . 6 | |
74 | 73 | 2rexbidv 2437 | . . . . 5 |
75 | oveq1 5749 | . . . . . . 7 | |
76 | 75 | eqeq2d 2129 | . . . . . 6 |
77 | oveq2 5750 | . . . . . . 7 | |
78 | 77 | eqeq2d 2129 | . . . . . 6 |
79 | 76, 78 | cbvrex2v 2640 | . . . . 5 |
80 | 74, 79 | syl6bb 195 | . . . 4 |
81 | 80 | cbvrabv 2659 | . . 3 |
82 | 73 | 2rexbidv 2437 | . . . . 5 |
83 | 76, 78 | cbvrex2v 2640 | . . . . 5 |
84 | 82, 83 | syl6bb 195 | . . . 4 |
85 | 84 | cbvrabv 2659 | . . 3 |
86 | 81, 85 | opeq12i 3680 | . 2 |
87 | 72, 86 | syl6eq 2166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 cab 2103 wrex 2394 crab 2397 cvv 2660 cop 3500 cxp 4507 cfv 5093 (class class class)co 5742 cmpo 5744 c1st 6004 c2nd 6005 cnq 7056 cnp 7067 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-qs 6403 df-ni 7080 df-nqqs 7124 df-inp 7242 |
This theorem is referenced by: genpelvl 7288 genpelvu 7289 plpvlu 7314 mpvlu 7315 |
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