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| Mirrors > Home > MPE Home > Th. List > dprddomcld | Structured version Visualization version GIF version | ||
| Description: If a family of subgroups is a family of subgroups for an internal direct product, then it is indexed by a set. (Contributed by AV, 13-Jul-2019.) |
| Ref | Expression |
|---|---|
| dprddomcld.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
| dprddomcld.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
| Ref | Expression |
|---|---|
| dprddomcld | ⊢ (𝜑 → 𝐼 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dprddomcld.2 | . 2 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
| 2 | dprddomcld.1 | . 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
| 3 | df-nel 3047 | . . . . 5 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
| 4 | dprddomprc 20020 | . . . . 5 ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆) | |
| 5 | 3, 4 | sylbir 235 | . . . 4 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝐺dom DProd 𝑆) |
| 6 | 5 | con4i 114 | . . 3 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
| 7 | eleq1 2829 | . . 3 ⊢ (dom 𝑆 = 𝐼 → (dom 𝑆 ∈ V ↔ 𝐼 ∈ V)) | |
| 8 | 6, 7 | imbitrid 244 | . 2 ⊢ (dom 𝑆 = 𝐼 → (𝐺dom DProd 𝑆 → 𝐼 ∈ V)) |
| 9 | 1, 2, 8 | sylc 65 | 1 ⊢ (𝜑 → 𝐼 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 Vcvv 3480 class class class wbr 5143 dom cdm 5685 DProd cdprd 20013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-oprab 7435 df-mpo 7436 df-dprd 20015 |
| This theorem is referenced by: dprdcntz 20028 dprddisj 20029 dprdw 20030 dprdwd 20031 dprdfid 20037 dprdfinv 20039 dprdfadd 20040 dprdfsub 20041 dprdfeq0 20042 dprdf11 20043 dprdlub 20046 dprdres 20048 dprdss 20049 dprdf1o 20052 dmdprdsplitlem 20057 dprddisj2 20059 dmdprdsplit2 20066 dpjfval 20075 dpjidcl 20078 |
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