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Theorem subdrgint 20821
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 20612 . . . 4 ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubRing‘𝑅))
41, 2, 3syl2anc 584 . . 3 (𝜑 𝑆 ∈ (SubRing‘𝑅))
5 subdrgint.1 . . . 4 𝐿 = (𝑅s 𝑆)
65subrgring 20591 . . 3 ( 𝑆 ∈ (SubRing‘𝑅) → 𝐿 ∈ Ring)
74, 6syl 17 . 2 (𝜑𝐿 ∈ Ring)
85fveq2i 6910 . . . 4 (mulGrp‘𝐿) = (mulGrp‘(𝑅s 𝑆))
98oveq1i 7441 . . 3 ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)}))
10 subdrgint.2 . . . . . . 7 (𝜑𝑅 ∈ DivRing)
11 eqid 2735 . . . . . . . 8 (𝑅s 𝑆) = (𝑅s 𝑆)
12 eqid 2735 . . . . . . . 8 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1311, 12mgpress 20167 . . . . . . 7 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → ((mulGrp‘𝑅) ↾s 𝑆) = (mulGrp‘(𝑅s 𝑆)))
1410, 4, 13syl2anc 584 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s 𝑆) = (mulGrp‘(𝑅s 𝑆)))
1514oveq1d 7446 . . . . 5 (𝜑 → (((mulGrp‘𝑅) ↾s 𝑆) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
16 difssd 4147 . . . . . . 7 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ (Base‘𝐿))
17 eqid 2735 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
1817subrgss 20589 . . . . . . . 8 ( 𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ (Base‘𝑅))
195, 17ressbas2 17283 . . . . . . . 8 ( 𝑆 ⊆ (Base‘𝑅) → 𝑆 = (Base‘𝐿))
204, 18, 193syl 18 . . . . . . 7 (𝜑 𝑆 = (Base‘𝐿))
2116, 20sseqtrrd 4037 . . . . . 6 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ 𝑆)
22 ressabs 17295 . . . . . 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 584 . . . . 5 (𝜑 → (((mulGrp‘𝑅) ↾s 𝑆) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
2415, 23eqtr3d 2777 . . . 4 (𝜑 → ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
25 intiin 5064 . . . . . . . 8 𝑆 = 𝑠𝑆 𝑠
2620, 25eqtr3di 2790 . . . . . . 7 (𝜑 → (Base‘𝐿) = 𝑠𝑆 𝑠)
2726difeq1d 4135 . . . . . 6 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
2827oveq2d 7447 . . . . 5 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})))
29 vex 3482 . . . . . . . . . 10 𝑠 ∈ V
3029difexi 5336 . . . . . . . . 9 (𝑠 ∖ {(0g𝐿)}) ∈ V
3130dfiin3 5984 . . . . . . . 8 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))
32 iindif1 5080 . . . . . . . . 9 (𝑆 ≠ ∅ → 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
332, 32syl 17 . . . . . . . 8 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
3431, 33eqtr3id 2789 . . . . . . 7 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
3534oveq2d 7447 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})))
36 difss 4146 . . . . . . . . . 10 ((Base‘𝑅) ∖ {(0g𝑅)}) ⊆ (Base‘𝑅)
37 eqid 2735 . . . . . . . . . . 11 ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))
3812, 17mgpbas 20158 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
3937, 38ressbas2 17283 . . . . . . . . . 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 6921 . . . . . . . 8 ((Base‘𝑅) ∖ {(0g𝑅)}) ∈ V
42 iinssiun 5010 . . . . . . . . . . 11 (𝑆 ≠ ∅ → 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}))
432, 42syl 17 . . . . . . . . . 10 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}))
44 subrgsubg 20594 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ∈ (SubGrp‘𝑅))
4544ssriv 3999 . . . . . . . . . . . . . . . . . 18 (SubRing‘𝑅) ⊆ (SubGrp‘𝑅)
461, 45sstrdi 4008 . . . . . . . . . . . . . . . . 17 (𝜑𝑆 ⊆ (SubGrp‘𝑅))
47 subgint 19181 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ (SubGrp‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubGrp‘𝑅))
4846, 2, 47syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 𝑆 ∈ (SubGrp‘𝑅))
49 eqid 2735 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
505, 49subg0 19163 . . . . . . . . . . . . . . . 16 ( 𝑆 ∈ (SubGrp‘𝑅) → (0g𝑅) = (0g𝐿))
5148, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0g𝑅) = (0g𝐿))
5251adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → (0g𝑅) = (0g𝐿))
5352sneqd 4643 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → {(0g𝑅)} = {(0g𝐿)})
5453difeq2d 4136 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) = (𝑠 ∖ {(0g𝐿)}))
551sselda 3995 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → 𝑠 ∈ (SubRing‘𝑅))
5617subrgss 20589 . . . . . . . . . . . . . 14 (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ⊆ (Base‘𝑅))
5755, 56syl 17 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → 𝑠 ⊆ (Base‘𝑅))
5857ssdifd 4155 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
5954, 58eqsstrrd 4035 . . . . . . . . . . 11 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6059iunssd 5055 . . . . . . . . . 10 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6143, 60sstrd 4006 . . . . . . . . 9 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6231, 61eqsstrrid 4045 . . . . . . . 8 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
63 ressabs 17295 . . . . . . . 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 587 . . . . . . 7 (𝜑 → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))))
6517, 49, 37drngmgp 20762 . . . . . . . . . . . . . 14 (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6610, 65syl 17 . . . . . . . . . . . . 13 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6766adantr 480 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6859, 40sseqtrdi 4046 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
69 ressabs 17295 . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
71 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝑅s 𝑠) = (𝑅s 𝑠)
7271, 12mgpress 20167 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ DivRing ∧ 𝑠𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅s 𝑠)))
7310, 72sylan 580 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅s 𝑠)))
7454eqcomd 2741 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) = (𝑠 ∖ {(0g𝑅)}))
7573, 74oveq12d 7449 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})))
76 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → 𝑠𝑆)
77 difssd 4147 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠)
78 ressabs 17295 . . . . . . . . . . . . . . . 16 ((𝑠𝑆 ∧ (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
7976, 77, 78syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
8075, 79eqtr3d 2777 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
8171, 17ressbas2 17283 . . . . . . . . . . . . . . . . . 18 (𝑠 ⊆ (Base‘𝑅) → 𝑠 = (Base‘(𝑅s 𝑠)))
8255, 56, 813syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠𝑆) → 𝑠 = (Base‘(𝑅s 𝑠)))
8371, 49subrg0 20596 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (SubRing‘𝑅) → (0g𝑅) = (0g‘(𝑅s 𝑠)))
8455, 83syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠𝑆) → (0g𝑅) = (0g‘(𝑅s 𝑠)))
8584sneqd 4643 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠𝑆) → {(0g𝑅)} = {(0g‘(𝑅s 𝑠))})
8682, 85difeq12d 4137 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) = ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))}))
8786oveq2d 7447 . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (Base‘(𝑅s 𝑠)) = (Base‘(𝑅s 𝑠))
90 eqid 2735 . . . . . . . . . . . . . . . . 17 (0g‘(𝑅s 𝑠)) = (0g‘(𝑅s 𝑠))
91 eqid 2735 . . . . . . . . . . . . . . . . 17 ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))})) = ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))}))
9289, 90, 91drngmgp 20762 . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) ∈ Grp)
9580, 94eqeltrrd 2840 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp)
9670, 95eqeltrd 2839 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp)
97 eqid 2735 . . . . . . . . . . . . 13 (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})))
9897issubg 19157 . . . . . . . . . . . 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 1342 . . . . . . . . . . 11 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
10099ralrimiva 3144 . . . . . . . . . 10 (𝜑 → ∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
101 eqid 2735 . . . . . . . . . . 11 (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))
102101rnmptss 7143 . . . . . . . . . 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 6264 . . . . . . . . . . . . 13 (∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ V → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆)
105 difexg 5335 . . . . . . . . . . . . 13 (𝑠𝑆 → (𝑠 ∖ {(0g𝐿)}) ∈ V)
106104, 105mprg 3065 . . . . . . . . . . . 12 dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆
107106a1i 11 . . . . . . . . . . 11 (𝜑 → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆)
108107, 2eqnetrd 3006 . . . . . . . . . 10 (𝜑 → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
109 dm0rn0 5938 . . . . . . . . . . 11 (dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ∅ ↔ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ∅)
110109necon3bii 2991 . . . . . . . . . 10 (dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅ ↔ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
111108, 110sylib 218 . . . . . . . . 9 (𝜑 → ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
112 subgint 19181 . . . . . . . . 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 584 . . . . . . . 8 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
114 eqid 2735 . . . . . . . . 9 (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})))
115114subggrp 19160 . . . . . . . 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 2840 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) ∈ Grp)
11835, 117eqeltrrd 2840 . . . . 5 (𝜑 → ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})) ∈ Grp)
11928, 118eqeltrd 2839 . . . 4 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
12024, 119eqeltrd 2839 . . 3 (𝜑 → ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
1219, 120eqeltrid 2843 . 2 (𝜑 → ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
122 eqid 2735 . . 3 (Base‘𝐿) = (Base‘𝐿)
123 eqid 2735 . . 3 (0g𝐿) = (0g𝐿)
124 eqid 2735 . . 3 ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)}))
125122, 123, 124isdrng2 20760 . 2 (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp))
1267, 121, 125sylanbrc 583 1 (𝜑𝐿 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  Vcvv 3478  cdif 3960  wss 3963  c0 4339  {csn 4631   cint 4951   ciun 4996   ciin 4997  cmpt 5231  dom cdm 5689  ran crn 5690  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  0gc0g 17486  Grpcgrp 18964  SubGrpcsubg 19151  mulGrpcmgp 20152  Ringcrg 20251  SubRingcsubrg 20586  DivRingcdr 20746
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-subrng 20563  df-subrg 20587  df-drng 20748
This theorem is referenced by:  sdrgint  20822  primefld  20823  fldgensdrg  33296
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