![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dprddisj | Structured version Visualization version GIF version |
Description: The function 𝑆 is a family having trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdcntz.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dprdcntz.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dprdcntz.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
dprddisj.0 | ⊢ 0 = (0g‘𝐺) |
dprddisj.k | ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) |
Ref | Expression |
---|---|
dprddisj | ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6889 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋)) | |
2 | sneq 4638 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
3 | 2 | difeq2d 4122 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋})) |
4 | 3 | imaeq2d 6058 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋}))) |
5 | 4 | unieqd 4922 | . . . . 5 ⊢ (𝑥 = 𝑋 → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑋}))) |
6 | 5 | fveq2d 6893 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) |
7 | 1, 6 | ineq12d 4213 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))))) |
8 | 7 | eqeq1d 2735 | . 2 ⊢ (𝑥 = 𝑋 → (((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 } ↔ ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })) |
9 | dprdcntz.1 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
10 | dprdcntz.2 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
11 | 9, 10 | dprddomcld 19866 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
12 | eqid 2733 | . . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
13 | dprddisj.0 | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
14 | dprddisj.k | . . . . . . 7 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
15 | 12, 13, 14 | dmdprd 19863 | . . . . . 6 ⊢ ((𝐼 ∈ V ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))) |
16 | 11, 10, 15 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))) |
17 | 9, 16 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))) |
18 | 17 | simp3d 1145 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })) |
19 | simpr 486 | . . . 4 ⊢ ((∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) | |
20 | 19 | ralimi 3084 | . . 3 ⊢ (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) → ∀𝑥 ∈ 𝐼 ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) |
21 | 18, 20 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }) |
22 | dprdcntz.3 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
23 | 8, 21, 22 | rspcdva 3614 | 1 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 {csn 4628 ∪ cuni 4908 class class class wbr 5148 dom cdm 5676 “ cima 5679 ⟶wf 6537 ‘cfv 6541 0gc0g 17382 mrClscmrc 17524 Grpcgrp 18816 SubGrpcsubg 18995 Cntzccntz 19174 DProd cdprd 19858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-oprab 7410 df-mpo 7411 df-1st 7972 df-2nd 7973 df-ixp 8889 df-dprd 19860 |
This theorem is referenced by: dprdfeq0 19887 dprdres 19893 dprdss 19894 dprdf1o 19897 dprd2da 19907 dmdprdsplit2lem 19910 |
Copyright terms: Public domain | W3C validator |