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Mirrors > Home > MPE Home > Th. List > dprdgrp | Structured version Visualization version GIF version |
Description: Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdgrp | ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldmdprd 19238 | . . . . . 6 ⊢ Rel dom DProd | |
2 | 1 | brrelex2i 5580 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
3 | 2 | dmexd 7636 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
4 | eqid 2738 | . . . 4 ⊢ dom 𝑆 = dom 𝑆 | |
5 | eqid 2738 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
6 | eqid 2738 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
7 | eqid 2738 | . . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | |
8 | 5, 6, 7 | dmdprd 19239 | . . . 4 ⊢ ((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)})))) |
9 | 3, 4, 8 | sylancl 589 | . . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)})))) |
10 | 9 | ibi 270 | . 2 ⊢ (𝐺dom DProd 𝑆 → (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)}))) |
11 | 10 | simp1d 1143 | 1 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3053 Vcvv 3398 ∖ cdif 3840 ∩ cin 3842 ⊆ wss 3843 {csn 4516 ∪ cuni 4796 class class class wbr 5030 dom cdm 5525 “ cima 5528 ⟶wf 6335 ‘cfv 6339 0gc0g 16816 mrClscmrc 16957 Grpcgrp 18219 SubGrpcsubg 18391 Cntzccntz 18563 DProd cdprd 19234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-ixp 8508 df-dprd 19236 |
This theorem is referenced by: dprdssv 19257 dprdfid 19258 dprdfinv 19260 dprdfadd 19261 dprdfsub 19262 dprdfeq0 19263 dprdf11 19264 dprdsubg 19265 dprdlub 19267 dprdspan 19268 dprdres 19269 dprdss 19270 dprdf1o 19273 dmdprdsplitlem 19278 dprdcntz2 19279 dprddisj2 19280 dprd2dlem1 19282 dprd2da 19283 dmdprdsplit2lem 19286 dmdprdsplit2 19287 dpjfval 19296 dpjidcl 19299 |
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