| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2736 | . . 3
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) | 
| 2 |  | eqid 2736 | . . 3
⊢
(0g‘𝐺) = (0g‘𝐺) | 
| 3 |  | eqid 2736 | . . 3
⊢
(mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | 
| 4 |  | dprdss.1 | . . . 4
⊢ (𝜑 → 𝐺dom DProd 𝑇) | 
| 5 |  | dprdgrp 20026 | . . . 4
⊢ (𝐺dom DProd 𝑇 → 𝐺 ∈ Grp) | 
| 6 | 4, 5 | syl 17 | . . 3
⊢ (𝜑 → 𝐺 ∈ Grp) | 
| 7 |  | dprdss.2 | . . . 4
⊢ (𝜑 → dom 𝑇 = 𝐼) | 
| 8 | 4, 7 | dprddomcld 20022 | . . 3
⊢ (𝜑 → 𝐼 ∈ V) | 
| 9 |  | dprdss.3 | . . 3
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) | 
| 10 |  | dprdss.4 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ (𝑇‘𝑘)) | 
| 11 | 10 | ralrimiva 3145 | . . . . . 6
⊢ (𝜑 → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘)) | 
| 12 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑘 = 𝑥 → (𝑆‘𝑘) = (𝑆‘𝑥)) | 
| 13 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥)) | 
| 14 | 12, 13 | sseq12d 4016 | . . . . . . 7
⊢ (𝑘 = 𝑥 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑥) ⊆ (𝑇‘𝑥))) | 
| 15 | 14 | rspcv 3617 | . . . . . 6
⊢ (𝑥 ∈ 𝐼 → (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥))) | 
| 16 | 11, 15 | mpan9 506 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥)) | 
| 17 | 16 | 3ad2antr1 1188 | . . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ (𝑇‘𝑥)) | 
| 18 | 4 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝐺dom DProd 𝑇) | 
| 19 | 7 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → dom 𝑇 = 𝐼) | 
| 20 |  | simpr1 1194 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐼) | 
| 21 |  | simpr2 1195 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐼) | 
| 22 |  | simpr3 1196 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦) | 
| 23 | 18, 19, 20, 21, 22, 1 | dprdcntz 20029 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑇‘𝑦))) | 
| 24 | 4, 7 | dprdf2 20028 | . . . . . . . . 9
⊢ (𝜑 → 𝑇:𝐼⟶(SubGrp‘𝐺)) | 
| 25 | 24 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → 𝑇:𝐼⟶(SubGrp‘𝐺)) | 
| 26 | 25, 21 | ffvelcdmd 7104 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑦) ∈ (SubGrp‘𝐺)) | 
| 27 |  | eqid 2736 | . . . . . . . 8
⊢
(Base‘𝐺) =
(Base‘𝐺) | 
| 28 | 27 | subgss 19146 | . . . . . . 7
⊢ ((𝑇‘𝑦) ∈ (SubGrp‘𝐺) → (𝑇‘𝑦) ⊆ (Base‘𝐺)) | 
| 29 | 26, 28 | syl 17 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑦) ⊆ (Base‘𝐺)) | 
| 30 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑘 = 𝑦 → (𝑆‘𝑘) = (𝑆‘𝑦)) | 
| 31 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦)) | 
| 32 | 30, 31 | sseq12d 4016 | . . . . . . 7
⊢ (𝑘 = 𝑦 → ((𝑆‘𝑘) ⊆ (𝑇‘𝑘) ↔ (𝑆‘𝑦) ⊆ (𝑇‘𝑦))) | 
| 33 | 11 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘)) | 
| 34 | 32, 33, 21 | rspcdva 3622 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑦) ⊆ (𝑇‘𝑦)) | 
| 35 | 27, 1 | cntz2ss 19354 | . . . . . 6
⊢ (((𝑇‘𝑦) ⊆ (Base‘𝐺) ∧ (𝑆‘𝑦) ⊆ (𝑇‘𝑦)) → ((Cntz‘𝐺)‘(𝑇‘𝑦)) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦))) | 
| 36 | 29, 34, 35 | syl2anc 584 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → ((Cntz‘𝐺)‘(𝑇‘𝑦)) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦))) | 
| 37 | 23, 36 | sstrd 3993 | . . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑇‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦))) | 
| 38 | 17, 37 | sstrd 3993 | . . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦))) | 
| 39 | 6 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp) | 
| 40 | 27 | subgacs 19180 | . . . . . . 7
⊢ (𝐺 ∈ Grp →
(SubGrp‘𝐺) ∈
(ACS‘(Base‘𝐺))) | 
| 41 |  | acsmre 17696 | . . . . . . 7
⊢
((SubGrp‘𝐺)
∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) | 
| 42 | 39, 40, 41 | 3syl 18 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) | 
| 43 |  | difss 4135 | . . . . . . . . 9
⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼 | 
| 44 | 11 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘)) | 
| 45 |  | ssralv 4051 | . . . . . . . . 9
⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → (∀𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → ∀𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘))) | 
| 46 | 43, 44, 45 | mpsyl 68 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘)) | 
| 47 |  | ss2iun 5009 | . . . . . . . 8
⊢
(∀𝑘 ∈
(𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ (𝑇‘𝑘) → ∪
𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ ∪
𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘)) | 
| 48 | 46, 47 | syl 17 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪
𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) ⊆ ∪
𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘)) | 
| 49 | 9 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺)) | 
| 50 |  | ffun 6738 | . . . . . . . 8
⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → Fun 𝑆) | 
| 51 |  | funiunfv 7269 | . . . . . . . 8
⊢ (Fun
𝑆 → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) = ∪ (𝑆 “ (𝐼 ∖ {𝑥}))) | 
| 52 | 49, 50, 51 | 3syl 18 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪
𝑘 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑘) = ∪ (𝑆 “ (𝐼 ∖ {𝑥}))) | 
| 53 | 24 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑇:𝐼⟶(SubGrp‘𝐺)) | 
| 54 |  | ffun 6738 | . . . . . . . 8
⊢ (𝑇:𝐼⟶(SubGrp‘𝐺) → Fun 𝑇) | 
| 55 |  | funiunfv 7269 | . . . . . . . 8
⊢ (Fun
𝑇 → ∪ 𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘) = ∪ (𝑇 “ (𝐼 ∖ {𝑥}))) | 
| 56 | 53, 54, 55 | 3syl 18 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪
𝑘 ∈ (𝐼 ∖ {𝑥})(𝑇‘𝑘) = ∪ (𝑇 “ (𝐼 ∖ {𝑥}))) | 
| 57 | 48, 52, 56 | 3sstr3d 4037 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ∪
(𝑇 “ (𝐼 ∖ {𝑥}))) | 
| 58 |  | imassrn 6088 | . . . . . . . 8
⊢ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑇 | 
| 59 | 53 | frnd 6743 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ (SubGrp‘𝐺)) | 
| 60 |  | mresspw 17636 | . . . . . . . . . 10
⊢
((SubGrp‘𝐺)
∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)) | 
| 61 | 42, 60 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)) | 
| 62 | 59, 61 | sstrd 3993 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ran 𝑇 ⊆ 𝒫 (Base‘𝐺)) | 
| 63 | 58, 62 | sstrid 3994 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺)) | 
| 64 |  | sspwuni 5099 | . . . . . . 7
⊢ ((𝑇 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑇
“ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) | 
| 65 | 63, 64 | sylib 218 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ (𝑇 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) | 
| 66 | 42, 3, 57, 65 | mrcssd 17668 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))) ⊆
((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))) | 
| 67 |  | ss2in 4244 | . . . . 5
⊢ (((𝑆‘𝑥) ⊆ (𝑇‘𝑥) ∧ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))) ⊆
((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇 “ (𝐼 ∖ {𝑥})))) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) ⊆ ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇
“ (𝐼 ∖ {𝑥}))))) | 
| 68 | 16, 66, 67 | syl2anc 584 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) ⊆ ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇
“ (𝐼 ∖ {𝑥}))))) | 
| 69 | 4 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑇) | 
| 70 | 7 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → dom 𝑇 = 𝐼) | 
| 71 |  | simpr 484 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | 
| 72 | 69, 70, 71, 2, 3 | dprddisj 20030 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑇‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑇
“ (𝐼 ∖ {𝑥})))) =
{(0g‘𝐺)}) | 
| 73 | 68, 72 | sseqtrd 4019 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) ⊆
{(0g‘𝐺)}) | 
| 74 | 1, 2, 3, 6, 8, 9, 38, 73 | dmdprdd 20020 | . 2
⊢ (𝜑 → 𝐺dom DProd 𝑆) | 
| 75 | 4 | a1d 25 | . . . . 5
⊢ (𝜑 → (𝐺dom DProd 𝑆 → 𝐺dom DProd 𝑇)) | 
| 76 |  | ss2ixp 8951 | . . . . . . 7
⊢
(∀𝑘 ∈
𝐼 (𝑆‘𝑘) ⊆ (𝑇‘𝑘) → X𝑘 ∈ 𝐼 (𝑆‘𝑘) ⊆ X𝑘 ∈ 𝐼 (𝑇‘𝑘)) | 
| 77 | 11, 76 | syl 17 | . . . . . 6
⊢ (𝜑 → X𝑘 ∈
𝐼 (𝑆‘𝑘) ⊆ X𝑘 ∈ 𝐼 (𝑇‘𝑘)) | 
| 78 |  | rabss2 4077 | . . . . . 6
⊢ (X𝑘 ∈
𝐼 (𝑆‘𝑘) ⊆ X𝑘 ∈ 𝐼 (𝑇‘𝑘) → {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} ⊆ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}) | 
| 79 |  | ssrexv 4052 | . . . . . 6
⊢ ({ℎ ∈ X𝑘 ∈
𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} ⊆ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} → (∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓) → ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓))) | 
| 80 | 77, 78, 79 | 3syl 18 | . . . . 5
⊢ (𝜑 → (∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓) → ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓))) | 
| 81 | 75, 80 | anim12d 609 | . . . 4
⊢ (𝜑 → ((𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)) → (𝐺dom DProd 𝑇 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)))) | 
| 82 |  | fdm 6744 | . . . . 5
⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → dom 𝑆 = 𝐼) | 
| 83 |  | eqid 2736 | . . . . . 6
⊢ {ℎ ∈ X𝑘 ∈
𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} | 
| 84 | 2, 83 | eldprd 20025 | . . . . 5
⊢ (dom
𝑆 = 𝐼 → (𝑎 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)))) | 
| 85 | 9, 82, 84 | 3syl 18 | . . . 4
⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑆‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)))) | 
| 86 |  | eqid 2736 | . . . . . 6
⊢ {ℎ ∈ X𝑘 ∈
𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)} | 
| 87 | 2, 86 | eldprd 20025 | . . . . 5
⊢ (dom
𝑇 = 𝐼 → (𝑎 ∈ (𝐺 DProd 𝑇) ↔ (𝐺dom DProd 𝑇 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)))) | 
| 88 | 7, 87 | syl 17 | . . . 4
⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑇) ↔ (𝐺dom DProd 𝑇 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑘 ∈ 𝐼 (𝑇‘𝑘) ∣ ℎ finSupp (0g‘𝐺)}𝑎 = (𝐺 Σg 𝑓)))) | 
| 89 | 81, 85, 88 | 3imtr4d 294 | . . 3
⊢ (𝜑 → (𝑎 ∈ (𝐺 DProd 𝑆) → 𝑎 ∈ (𝐺 DProd 𝑇))) | 
| 90 | 89 | ssrdv 3988 | . 2
⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇)) | 
| 91 | 74, 90 | jca 511 | 1
⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇))) |