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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 3045 | . . . . 5 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
4 | dprddomprc 20035 | . . . . 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 2827 | . . 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 1537 ∈ wcel 2106 ∉ wnel 3044 Vcvv 3478 class class class wbr 5148 dom cdm 5689 DProd cdprd 20028 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-oprab 7435 df-mpo 7436 df-dprd 20030 |
This theorem is referenced by: dprdcntz 20043 dprddisj 20044 dprdw 20045 dprdwd 20046 dprdfid 20052 dprdfinv 20054 dprdfadd 20055 dprdfsub 20056 dprdfeq0 20057 dprdf11 20058 dprdlub 20061 dprdres 20063 dprdss 20064 dprdf1o 20067 dmdprdsplitlem 20072 dprddisj2 20074 dmdprdsplit2 20081 dpjfval 20090 dpjidcl 20093 |
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