| 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 6834 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋)) | |
| 2 | sneq 4590 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
| 3 | 2 | difeq2d 4078 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋})) |
| 4 | 3 | imaeq2d 6019 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 “ (𝐼 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑋}))) |
| 5 | 4 | unieqd 4876 | . . . . 5 ⊢ (𝑥 = 𝑋 → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑋}))) |
| 6 | 5 | fveq2d 6838 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) |
| 7 | 1, 6 | ineq12d 4173 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))))) |
| 8 | 7 | eqeq1d 2738 | . 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 19932 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
| 12 | eqid 2736 | . . . . . . 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 19929 | . . . . . 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 584 | . . . . 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 232 | . . . 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 1144 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })) |
| 19 | simpr 484 | . . . 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 3073 | . . 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 3577 | 1 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3051 Vcvv 3440 ∖ cdif 3898 ∩ cin 3900 ⊆ wss 3901 {csn 4580 ∪ cuni 4863 class class class wbr 5098 dom cdm 5624 “ cima 5627 ⟶wf 6488 ‘cfv 6492 0gc0g 17359 mrClscmrc 17502 Grpcgrp 18863 SubGrpcsubg 19050 Cntzccntz 19244 DProd cdprd 19924 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-ixp 8836 df-dprd 19926 |
| This theorem is referenced by: dprdfeq0 19953 dprdres 19959 dprdss 19960 dprdf1o 19963 dprd2da 19973 dmdprdsplit2lem 19976 |
| Copyright terms: Public domain | W3C validator |