MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subdrgint Structured version   Visualization version   GIF version

Theorem subdrgint 20778
Description: The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
Hypotheses
Ref Expression
subdrgint.1 𝐿 = (𝑅s 𝑆)
subdrgint.2 (𝜑𝑅 ∈ DivRing)
subdrgint.3 (𝜑𝑆 ⊆ (SubRing‘𝑅))
subdrgint.4 (𝜑𝑆 ≠ ∅)
subdrgint.5 ((𝜑𝑠𝑆) → (𝑅s 𝑠) ∈ DivRing)
Assertion
Ref Expression
subdrgint (𝜑𝐿 ∈ DivRing)
Distinct variable groups:   𝐿,𝑠   𝑅,𝑠   𝑆,𝑠   𝜑,𝑠

Proof of Theorem subdrgint
StepHypRef Expression
1 subdrgint.3 . . . 4 (𝜑𝑆 ⊆ (SubRing‘𝑅))
2 subdrgint.4 . . . 4 (𝜑𝑆 ≠ ∅)
3 subrgint 20575 . . . 4 ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubRing‘𝑅))
41, 2, 3syl2anc 582 . . 3 (𝜑 𝑆 ∈ (SubRing‘𝑅))
5 subdrgint.1 . . . 4 𝐿 = (𝑅s 𝑆)
65subrgring 20554 . . 3 ( 𝑆 ∈ (SubRing‘𝑅) → 𝐿 ∈ Ring)
74, 6syl 17 . 2 (𝜑𝐿 ∈ Ring)
85fveq2i 6896 . . . 4 (mulGrp‘𝐿) = (mulGrp‘(𝑅s 𝑆))
98oveq1i 7426 . . 3 ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)}))
10 subdrgint.2 . . . . . . 7 (𝜑𝑅 ∈ DivRing)
11 eqid 2726 . . . . . . . 8 (𝑅s 𝑆) = (𝑅s 𝑆)
12 eqid 2726 . . . . . . . 8 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1311, 12mgpress 20128 . . . . . . 7 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → ((mulGrp‘𝑅) ↾s 𝑆) = (mulGrp‘(𝑅s 𝑆)))
1410, 4, 13syl2anc 582 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s 𝑆) = (mulGrp‘(𝑅s 𝑆)))
1514oveq1d 7431 . . . . 5 (𝜑 → (((mulGrp‘𝑅) ↾s 𝑆) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
16 difssd 4129 . . . . . . 7 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ (Base‘𝐿))
17 eqid 2726 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
1817subrgss 20552 . . . . . . . 8 ( 𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ (Base‘𝑅))
195, 17ressbas2 17246 . . . . . . . 8 ( 𝑆 ⊆ (Base‘𝑅) → 𝑆 = (Base‘𝐿))
204, 18, 193syl 18 . . . . . . 7 (𝜑 𝑆 = (Base‘𝐿))
2116, 20sseqtrrd 4020 . . . . . 6 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ 𝑆)
22 ressabs 17258 . . . . . 6 (( 𝑆 ∈ (SubRing‘𝑅) ∧ ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ 𝑆) → (((mulGrp‘𝑅) ↾s 𝑆) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
234, 21, 22syl2anc 582 . . . . 5 (𝜑 → (((mulGrp‘𝑅) ↾s 𝑆) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
2415, 23eqtr3d 2768 . . . 4 (𝜑 → ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
25 intiin 5059 . . . . . . . 8 𝑆 = 𝑠𝑆 𝑠
2620, 25eqtr3di 2781 . . . . . . 7 (𝜑 → (Base‘𝐿) = 𝑠𝑆 𝑠)
2726difeq1d 4117 . . . . . 6 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
2827oveq2d 7432 . . . . 5 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})))
29 vex 3466 . . . . . . . . . 10 𝑠 ∈ V
3029difexi 5327 . . . . . . . . 9 (𝑠 ∖ {(0g𝐿)}) ∈ V
3130dfiin3 5966 . . . . . . . 8 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))
32 iindif1 5075 . . . . . . . . 9 (𝑆 ≠ ∅ → 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
332, 32syl 17 . . . . . . . 8 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
3431, 33eqtr3id 2780 . . . . . . 7 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
3534oveq2d 7432 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})))
36 difss 4128 . . . . . . . . . 10 ((Base‘𝑅) ∖ {(0g𝑅)}) ⊆ (Base‘𝑅)
37 eqid 2726 . . . . . . . . . . 11 ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))
3812, 17mgpbas 20119 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
3937, 38ressbas2 17246 . . . . . . . . . 10 (((Base‘𝑅) ∖ {(0g𝑅)}) ⊆ (Base‘𝑅) → ((Base‘𝑅) ∖ {(0g𝑅)}) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
4036, 39ax-mp 5 . . . . . . . . 9 ((Base‘𝑅) ∖ {(0g𝑅)}) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})))
4140fvexi 6907 . . . . . . . 8 ((Base‘𝑅) ∖ {(0g𝑅)}) ∈ V
42 iinssiun 5006 . . . . . . . . . . 11 (𝑆 ≠ ∅ → 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}))
432, 42syl 17 . . . . . . . . . 10 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}))
44 subrgsubg 20557 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ∈ (SubGrp‘𝑅))
4544ssriv 3982 . . . . . . . . . . . . . . . . . 18 (SubRing‘𝑅) ⊆ (SubGrp‘𝑅)
461, 45sstrdi 3991 . . . . . . . . . . . . . . . . 17 (𝜑𝑆 ⊆ (SubGrp‘𝑅))
47 subgint 19140 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ (SubGrp‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubGrp‘𝑅))
4846, 2, 47syl2anc 582 . . . . . . . . . . . . . . . 16 (𝜑 𝑆 ∈ (SubGrp‘𝑅))
49 eqid 2726 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
505, 49subg0 19122 . . . . . . . . . . . . . . . 16 ( 𝑆 ∈ (SubGrp‘𝑅) → (0g𝑅) = (0g𝐿))
5148, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0g𝑅) = (0g𝐿))
5251adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → (0g𝑅) = (0g𝐿))
5352sneqd 4635 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → {(0g𝑅)} = {(0g𝐿)})
5453difeq2d 4118 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) = (𝑠 ∖ {(0g𝐿)}))
551sselda 3978 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → 𝑠 ∈ (SubRing‘𝑅))
5617subrgss 20552 . . . . . . . . . . . . . 14 (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ⊆ (Base‘𝑅))
5755, 56syl 17 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → 𝑠 ⊆ (Base‘𝑅))
5857ssdifd 4137 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
5954, 58eqsstrrd 4018 . . . . . . . . . . 11 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6059iunssd 5050 . . . . . . . . . 10 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6143, 60sstrd 3989 . . . . . . . . 9 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6231, 61eqsstrrid 4028 . . . . . . . 8 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
63 ressabs 17258 . . . . . . . 8 ((((Base‘𝑅) ∖ {(0g𝑅)}) ∈ V ∧ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)})) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))))
6441, 62, 63sylancr 585 . . . . . . 7 (𝜑 → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))))
6517, 49, 37drngmgp 20719 . . . . . . . . . . . . . 14 (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6610, 65syl 17 . . . . . . . . . . . . 13 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6766adantr 479 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6859, 40sseqtrdi 4029 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
69 ressabs 17258 . . . . . . . . . . . . . 14 ((((Base‘𝑅) ∖ {(0g𝑅)}) ∈ V ∧ (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)})) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
7041, 59, 69sylancr 585 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
71 eqid 2726 . . . . . . . . . . . . . . . . . 18 (𝑅s 𝑠) = (𝑅s 𝑠)
7271, 12mgpress 20128 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ DivRing ∧ 𝑠𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅s 𝑠)))
7310, 72sylan 578 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅s 𝑠)))
7454eqcomd 2732 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) = (𝑠 ∖ {(0g𝑅)}))
7573, 74oveq12d 7434 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})))
76 simpr 483 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → 𝑠𝑆)
77 difssd 4129 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠)
78 ressabs 17258 . . . . . . . . . . . . . . . 16 ((𝑠𝑆 ∧ (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
7976, 77, 78syl2anc 582 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
8075, 79eqtr3d 2768 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
8171, 17ressbas2 17246 . . . . . . . . . . . . . . . . . 18 (𝑠 ⊆ (Base‘𝑅) → 𝑠 = (Base‘(𝑅s 𝑠)))
8255, 56, 813syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠𝑆) → 𝑠 = (Base‘(𝑅s 𝑠)))
8371, 49subrg0 20559 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (SubRing‘𝑅) → (0g𝑅) = (0g‘(𝑅s 𝑠)))
8455, 83syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠𝑆) → (0g𝑅) = (0g‘(𝑅s 𝑠)))
8584sneqd 4635 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠𝑆) → {(0g𝑅)} = {(0g‘(𝑅s 𝑠))})
8682, 85difeq12d 4119 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) = ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))}))
8786oveq2d 7432 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) = ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))})))
88 subdrgint.5 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑅s 𝑠) ∈ DivRing)
89 eqid 2726 . . . . . . . . . . . . . . . . 17 (Base‘(𝑅s 𝑠)) = (Base‘(𝑅s 𝑠))
90 eqid 2726 . . . . . . . . . . . . . . . . 17 (0g‘(𝑅s 𝑠)) = (0g‘(𝑅s 𝑠))
91 eqid 2726 . . . . . . . . . . . . . . . . 17 ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))})) = ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))}))
9289, 90, 91drngmgp 20719 . . . . . . . . . . . . . . . 16 ((𝑅s 𝑠) ∈ DivRing → ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))})) ∈ Grp)
9388, 92syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))})) ∈ Grp)
9487, 93eqeltrd 2826 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) ∈ Grp)
9580, 94eqeltrrd 2827 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp)
9670, 95eqeltrd 2826 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp)
97 eqid 2726 . . . . . . . . . . . . 13 (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})))
9897issubg 19116 . . . . . . . . . . . 12 ((𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) ↔ (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp ∧ (𝑠 ∖ {(0g𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) ∧ (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp))
9967, 68, 96, 98syl3anbrc 1340 . . . . . . . . . . 11 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
10099ralrimiva 3136 . . . . . . . . . 10 (𝜑 → ∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
101 eqid 2726 . . . . . . . . . . 11 (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))
102101rnmptss 7129 . . . . . . . . . 10 (∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) → ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ⊆ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
103100, 102syl 17 . . . . . . . . 9 (𝜑 → ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ⊆ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
104 dmmptg 6245 . . . . . . . . . . . . 13 (∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ V → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆)
105 difexg 5326 . . . . . . . . . . . . 13 (𝑠𝑆 → (𝑠 ∖ {(0g𝐿)}) ∈ V)
106104, 105mprg 3057 . . . . . . . . . . . 12 dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆
107106a1i 11 . . . . . . . . . . 11 (𝜑 → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆)
108107, 2eqnetrd 2998 . . . . . . . . . 10 (𝜑 → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
109 dm0rn0 5923 . . . . . . . . . . 11 (dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ∅ ↔ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ∅)
110109necon3bii 2983 . . . . . . . . . 10 (dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅ ↔ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
111108, 110sylib 217 . . . . . . . . 9 (𝜑 → ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
112 subgint 19140 . . . . . . . . 9 ((ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ⊆ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) ∧ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅) → ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
113103, 111, 112syl2anc 582 . . . . . . . 8 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
114 eqid 2726 . . . . . . . . 9 (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})))
115114subggrp 19119 . . . . . . . 8 ( ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) ∈ Grp)
116113, 115syl 17 . . . . . . 7 (𝜑 → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) ∈ Grp)
11764, 116eqeltrrd 2827 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) ∈ Grp)
11835, 117eqeltrrd 2827 . . . . 5 (𝜑 → ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})) ∈ Grp)
11928, 118eqeltrd 2826 . . . 4 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
12024, 119eqeltrd 2826 . . 3 (𝜑 → ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
1219, 120eqeltrid 2830 . 2 (𝜑 → ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
122 eqid 2726 . . 3 (Base‘𝐿) = (Base‘𝐿)
123 eqid 2726 . . 3 (0g𝐿) = (0g𝐿)
124 eqid 2726 . . 3 ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)}))
125122, 123, 124isdrng2 20717 . 2 (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp))
1267, 121, 125sylanbrc 581 1 (𝜑𝐿 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  Vcvv 3462  cdif 3943  wss 3946  c0 4322  {csn 4623   cint 4946   ciun 4993   ciin 4994  cmpt 5228  dom cdm 5674  ran crn 5675  cfv 6546  (class class class)co 7416  Basecbs 17208  s cress 17237  0gc0g 17449  Grpcgrp 18923  SubGrpcsubg 19110  mulGrpcmgp 20113  Ringcrg 20212  SubRingcsubrg 20547  DivRingcdr 20703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-tpos 8233  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-3 12322  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-mulr 17275  df-0g 17451  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-grp 18926  df-minusg 18927  df-subg 19113  df-cmn 19776  df-abl 19777  df-mgp 20114  df-rng 20132  df-ur 20161  df-ring 20214  df-oppr 20312  df-dvdsr 20335  df-unit 20336  df-invr 20366  df-dvr 20379  df-subrng 20524  df-subrg 20549  df-drng 20705
This theorem is referenced by:  sdrgint  20779  primefld  20780  fldgensdrg  33169
  Copyright terms: Public domain W3C validator