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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 6664 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋)) | |
2 | sneq 4570 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
3 | 2 | difeq2d 4098 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋})) |
4 | 3 | imaeq2d 5923 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋}))) |
5 | 4 | unieqd 4841 | . . . . 5 ⊢ (𝑥 = 𝑋 → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑋}))) |
6 | 5 | fveq2d 6668 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) |
7 | 1, 6 | ineq12d 4189 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))))) |
8 | 7 | eqeq1d 2823 | . 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 19117 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
12 | eqid 2821 | . . . . . . 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 19114 | . . . . . 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 586 | . . . . 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 234 | . . . 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 1140 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })) |
19 | simpr 487 | . . . 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 3160 | . . 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 3624 | 1 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 {csn 4560 ∪ cuni 4831 class class class wbr 5058 dom cdm 5549 “ cima 5552 ⟶wf 6345 ‘cfv 6349 0gc0g 16707 mrClscmrc 16848 Grpcgrp 18097 SubGrpcsubg 18267 Cntzccntz 18439 DProd cdprd 19109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-ixp 8456 df-dprd 19111 |
This theorem is referenced by: dprdfeq0 19138 dprdres 19144 dprdss 19145 dprdf1o 19148 dprd2da 19158 dmdprdsplit2lem 19161 |
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