Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dprdf2 | Structured version Visualization version GIF version |
Description: The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdcntz.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dprdcntz.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
Ref | Expression |
---|---|
dprdf2 | ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprdcntz.1 | . . 3 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dprdf 19057 | . . 3 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺)) |
4 | dprdcntz.2 | . . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
5 | 4 | feq2d 6493 | . 2 ⊢ (𝜑 → (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺))) |
6 | 3, 5 | mpbid 233 | 1 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 class class class wbr 5057 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 SubGrpcsubg 18211 DProd cdprd 19044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-ixp 8450 df-dprd 19046 |
This theorem is referenced by: dprdff 19063 dprdfid 19068 dprdfinv 19070 dprdfadd 19071 dprdfeq0 19073 dprdres 19079 dprdss 19080 dprdf1o 19083 dprdf1 19084 subgdprd 19086 dmdprdsplitlem 19088 dprdcntz2 19089 dpjlem 19102 dpjcntz 19103 dpjdisj 19104 dpjlsm 19105 dpjf 19108 dpjidcl 19109 dpjlid 19112 dpjghm 19114 dpjghm2 19115 ablfac1c 19122 ablfac1eulem 19123 ablfac1eu 19124 ablfaclem2 19137 ablfaclem3 19138 dchrptlem3 25769 |
Copyright terms: Public domain | W3C validator |