Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5539 | . . . 4 | |
2 | fveq2 5198 | . . . . . . 7 | |
3 | 2 | rexeqdv 2556 | . . . . . 6 |
4 | 3 | rabbidv 2593 | . . . . 5 |
5 | fveq2 5198 | . . . . . . 7 | |
6 | 5 | rexeqdv 2556 | . . . . . 6 |
7 | 6 | rabbidv 2593 | . . . . 5 |
8 | 4, 7 | opeq12d 3578 | . . . 4 |
9 | 1, 8 | eqeq12d 2095 | . . 3 |
10 | oveq2 5540 | . . . 4 | |
11 | fveq2 5198 | . . . . . . . 8 | |
12 | 11 | rexeqdv 2556 | . . . . . . 7 |
13 | 12 | rexbidv 2369 | . . . . . 6 |
14 | 13 | rabbidv 2593 | . . . . 5 |
15 | fveq2 5198 | . . . . . . . 8 | |
16 | 15 | rexeqdv 2556 | . . . . . . 7 |
17 | 16 | rexbidv 2369 | . . . . . 6 |
18 | 17 | rabbidv 2593 | . . . . 5 |
19 | 14, 18 | opeq12d 3578 | . . . 4 |
20 | 10, 19 | eqeq12d 2095 | . . 3 |
21 | nqex 6553 | . . . . . . 7 | |
22 | 21 | a1i 9 | . . . . . 6 |
23 | rabssab 3081 | . . . . . . 7 | |
24 | prop 6665 | . . . . . . . . . . . 12 | |
25 | elprnql 6671 | . . . . . . . . . . . 12 | |
26 | 24, 25 | sylan 277 | . . . . . . . . . . 11 |
27 | prop 6665 | . . . . . . . . . . . 12 | |
28 | elprnql 6671 | . . . . . . . . . . . 12 | |
29 | 27, 28 | sylan 277 | . . . . . . . . . . 11 |
30 | genp.2 | . . . . . . . . . . . 12 | |
31 | eleq1 2141 | . . . . . . . . . . . 12 | |
32 | 30, 31 | syl5ibrcom 155 | . . . . . . . . . . 11 |
33 | 26, 29, 32 | syl2an 283 | . . . . . . . . . 10 |
34 | 33 | an4s 552 | . . . . . . . . 9 |
35 | 34 | rexlimdvva 2484 | . . . . . . . 8 |
36 | 35 | abssdv 3068 | . . . . . . 7 |
37 | 23, 36 | syl5ss 3010 | . . . . . 6 |
38 | 22, 37 | ssexd 3918 | . . . . 5 |
39 | rabssab 3081 | . . . . . . 7 | |
40 | elprnqu 6672 | . . . . . . . . . . . 12 | |
41 | 24, 40 | sylan 277 | . . . . . . . . . . 11 |
42 | elprnqu 6672 | . . . . . . . . . . . 12 | |
43 | 27, 42 | sylan 277 | . . . . . . . . . . 11 |
44 | 41, 43, 32 | syl2an 283 | . . . . . . . . . 10 |
45 | 44 | an4s 552 | . . . . . . . . 9 |
46 | 45 | rexlimdvva 2484 | . . . . . . . 8 |
47 | 46 | abssdv 3068 | . . . . . . 7 |
48 | 39, 47 | syl5ss 3010 | . . . . . 6 |
49 | 22, 48 | ssexd 3918 | . . . . 5 |
50 | opelxp 4392 | . . . . 5 | |
51 | 38, 49, 50 | sylanbrc 408 | . . . 4 |
52 | fveq2 5198 | . . . . . . . 8 | |
53 | 52 | rexeqdv 2556 | . . . . . . 7 |
54 | 53 | rabbidv 2593 | . . . . . 6 |
55 | fveq2 5198 | . . . . . . . 8 | |
56 | 55 | rexeqdv 2556 | . . . . . . 7 |
57 | 56 | rabbidv 2593 | . . . . . 6 |
58 | 54, 57 | opeq12d 3578 | . . . . 5 |
59 | fveq2 5198 | . . . . . . . . 9 | |
60 | 59 | rexeqdv 2556 | . . . . . . . 8 |
61 | 60 | rexbidv 2369 | . . . . . . 7 |
62 | 61 | rabbidv 2593 | . . . . . 6 |
63 | fveq2 5198 | . . . . . . . . 9 | |
64 | 63 | rexeqdv 2556 | . . . . . . . 8 |
65 | 64 | rexbidv 2369 | . . . . . . 7 |
66 | 65 | rabbidv 2593 | . . . . . 6 |
67 | 62, 66 | opeq12d 3578 | . . . . 5 |
68 | genp.1 | . . . . . 6 | |
69 | 68 | genpdf 6698 | . . . . 5 |
70 | 58, 67, 69 | ovmpt2g 5655 | . . . 4 |
71 | 51, 70 | mpd3an3 1269 | . . 3 |
72 | 9, 20, 71 | vtocl2ga 2666 | . 2 |
73 | eqeq1 2087 | . . . . . 6 | |
74 | 73 | 2rexbidv 2391 | . . . . 5 |
75 | oveq1 5539 | . . . . . . 7 | |
76 | 75 | eqeq2d 2092 | . . . . . 6 |
77 | oveq2 5540 | . . . . . . 7 | |
78 | 77 | eqeq2d 2092 | . . . . . 6 |
79 | 76, 78 | cbvrex2v 2586 | . . . . 5 |
80 | 74, 79 | syl6bb 194 | . . . 4 |
81 | 80 | cbvrabv 2600 | . . 3 |
82 | 73 | 2rexbidv 2391 | . . . . 5 |
83 | 76, 78 | cbvrex2v 2586 | . . . . 5 |
84 | 82, 83 | syl6bb 194 | . . . 4 |
85 | 84 | cbvrabv 2600 | . . 3 |
86 | 81, 85 | opeq12i 3575 | . 2 |
87 | 72, 86 | syl6eq 2129 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 cab 2067 wrex 2349 crab 2352 cvv 2601 cop 3401 cxp 4361 cfv 4922 (class class class)co 5532 cmpt2 5534 c1st 5785 c2nd 5786 cnq 6470 cnp 6481 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-qs 6135 df-ni 6494 df-nqqs 6538 df-inp 6656 |
This theorem is referenced by: genpelvl 6702 genpelvu 6703 plpvlu 6728 mpvlu 6729 |
Copyright terms: Public domain | W3C validator |