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Theorem subdrgint 19272
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 19247 . . . 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 19228 . . 3 ( 𝑆 ∈ (SubRing‘𝑅) → 𝐿 ∈ Ring)
74, 6syl 17 . 2 (𝜑𝐿 ∈ Ring)
85fveq2i 6541 . . . 4 (mulGrp‘𝐿) = (mulGrp‘(𝑅s 𝑆))
98oveq1i 7026 . . 3 ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)}))
10 subdrgint.2 . . . . . . 7 (𝜑𝑅 ∈ DivRing)
11 eqid 2795 . . . . . . . 8 (𝑅s 𝑆) = (𝑅s 𝑆)
12 eqid 2795 . . . . . . . 8 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1311, 12mgpress 18940 . . . . . . 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 7031 . . . . 5 (𝜑 → (((mulGrp‘𝑅) ↾s 𝑆) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
16 difssd 4030 . . . . . . 7 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ (Base‘𝐿))
17 eqid 2795 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
1817subrgss 19226 . . . . . . . 8 ( 𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ (Base‘𝑅))
195, 17ressbas2 16384 . . . . . . . 8 ( 𝑆 ⊆ (Base‘𝑅) → 𝑆 = (Base‘𝐿))
204, 18, 193syl 18 . . . . . . 7 (𝜑 𝑆 = (Base‘𝐿))
2116, 20sseqtr4d 3929 . . . . . 6 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ 𝑆)
22 ressabs 16392 . . . . . 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 2833 . . . 4 (𝜑 → ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
25 intiin 4882 . . . . . . . 8 𝑆 = 𝑠𝑆 𝑠
2625, 20syl5reqr 2846 . . . . . . 7 (𝜑 → (Base‘𝐿) = 𝑠𝑆 𝑠)
2726difeq1d 4019 . . . . . 6 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
2827oveq2d 7032 . . . . 5 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})))
29 vex 3440 . . . . . . . . . 10 𝑠 ∈ V
3029difexi 5123 . . . . . . . . 9 (𝑠 ∖ {(0g𝐿)}) ∈ V
3130dfiin3 5719 . . . . . . . 8 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))
32 iindif1 4896 . . . . . . . . 9 (𝑆 ≠ ∅ → 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
332, 32syl 17 . . . . . . . 8 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
3431, 33syl5eqr 2845 . . . . . . 7 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
3534oveq2d 7032 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})))
36 difss 4029 . . . . . . . . . 10 ((Base‘𝑅) ∖ {(0g𝑅)}) ⊆ (Base‘𝑅)
37 eqid 2795 . . . . . . . . . . 11 ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))
3812, 17mgpbas 18935 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
3937, 38ressbas2 16384 . . . . . . . . . 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 6552 . . . . . . . 8 ((Base‘𝑅) ∖ {(0g𝑅)}) ∈ V
42 iinssiun 4837 . . . . . . . . . . 11 (𝑆 ≠ ∅ → 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}))
432, 42syl 17 . . . . . . . . . 10 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}))
44 subrgsubg 19231 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ∈ (SubGrp‘𝑅))
4544ssriv 3893 . . . . . . . . . . . . . . . . . 18 (SubRing‘𝑅) ⊆ (SubGrp‘𝑅)
461, 45syl6ss 3901 . . . . . . . . . . . . . . . . 17 (𝜑𝑆 ⊆ (SubGrp‘𝑅))
47 subgint 18057 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ (SubGrp‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubGrp‘𝑅))
4846, 2, 47syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 𝑆 ∈ (SubGrp‘𝑅))
49 eqid 2795 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
505, 49subg0 18039 . . . . . . . . . . . . . . . 16 ( 𝑆 ∈ (SubGrp‘𝑅) → (0g𝑅) = (0g𝐿))
5148, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0g𝑅) = (0g𝐿))
5251adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → (0g𝑅) = (0g𝐿))
5352sneqd 4484 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → {(0g𝑅)} = {(0g𝐿)})
5453difeq2d 4020 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) = (𝑠 ∖ {(0g𝐿)}))
551sselda 3889 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → 𝑠 ∈ (SubRing‘𝑅))
5617subrgss 19226 . . . . . . . . . . . . . 14 (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ⊆ (Base‘𝑅))
5755, 56syl 17 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → 𝑠 ⊆ (Base‘𝑅))
5857ssdifd 4038 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
5954, 58eqsstrrd 3927 . . . . . . . . . . 11 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6059iunssd 4873 . . . . . . . . . 10 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6143, 60sstrd 3899 . . . . . . . . 9 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6231, 61eqsstrrid 3937 . . . . . . . 8 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
63 ressabs 16392 . . . . . . . 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 19204 . . . . . . . . . . . . . 14 (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6610, 65syl 17 . . . . . . . . . . . . 13 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6766adantr 481 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6859, 40syl6sseq 3938 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
69 ressabs 16392 . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . . 18 (𝑅s 𝑠) = (𝑅s 𝑠)
7271, 12mgpress 18940 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ DivRing ∧ 𝑠𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅s 𝑠)))
7310, 72sylan 580 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅s 𝑠)))
7454eqcomd 2801 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) = (𝑠 ∖ {(0g𝑅)}))
7573, 74oveq12d 7034 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})))
76 simpr 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → 𝑠𝑆)
77 difssd 4030 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠)
78 ressabs 16392 . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
8171, 17ressbas2 16384 . . . . . . . . . . . . . . . . . 18 (𝑠 ⊆ (Base‘𝑅) → 𝑠 = (Base‘(𝑅s 𝑠)))
8255, 56, 813syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠𝑆) → 𝑠 = (Base‘(𝑅s 𝑠)))
8371, 49subrg0 19232 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (SubRing‘𝑅) → (0g𝑅) = (0g‘(𝑅s 𝑠)))
8455, 83syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠𝑆) → (0g𝑅) = (0g‘(𝑅s 𝑠)))
8584sneqd 4484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠𝑆) → {(0g𝑅)} = {(0g‘(𝑅s 𝑠))})
8682, 85difeq12d 4021 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) = ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))}))
8786oveq2d 7032 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . 17 (Base‘(𝑅s 𝑠)) = (Base‘(𝑅s 𝑠))
90 eqid 2795 . . . . . . . . . . . . . . . . 17 (0g‘(𝑅s 𝑠)) = (0g‘(𝑅s 𝑠))
91 eqid 2795 . . . . . . . . . . . . . . . . 17 ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))})) = ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))}))
9289, 90, 91drngmgp 19204 . . . . . . . . . . . . . . . 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 2883 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) ∈ Grp)
9580, 94eqeltrrd 2884 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp)
9670, 95eqeltrd 2883 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp)
97 eqid 2795 . . . . . . . . . . . . 13 (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})))
9897issubg 18033 . . . . . . . . . . . 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 1336 . . . . . . . . . . 11 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
10099ralrimiva 3149 . . . . . . . . . 10 (𝜑 → ∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
101 eqid 2795 . . . . . . . . . . 11 (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))
102101rnmptss 6749 . . . . . . . . . 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 5971 . . . . . . . . . . . . 13 (∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ V → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆)
105 difexg 5122 . . . . . . . . . . . . 13 (𝑠𝑆 → (𝑠 ∖ {(0g𝐿)}) ∈ V)
106104, 105mprg 3119 . . . . . . . . . . . 12 dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆
107106a1i 11 . . . . . . . . . . 11 (𝜑 → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆)
108107, 2eqnetrd 3051 . . . . . . . . . 10 (𝜑 → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
109 dm0rn0 5679 . . . . . . . . . . 11 (dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ∅ ↔ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ∅)
110109necon3bii 3036 . . . . . . . . . 10 (dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅ ↔ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
111108, 110sylib 219 . . . . . . . . 9 (𝜑 → ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
112 subgint 18057 . . . . . . . . 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 2795 . . . . . . . . 9 (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})))
115114subggrp 18036 . . . . . . . 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 2884 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) ∈ Grp)
11835, 117eqeltrrd 2884 . . . . 5 (𝜑 → ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})) ∈ Grp)
11928, 118eqeltrd 2883 . . . 4 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
12024, 119eqeltrd 2883 . . 3 (𝜑 → ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
1219, 120syl5eqel 2887 . 2 (𝜑 → ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
122 eqid 2795 . . 3 (Base‘𝐿) = (Base‘𝐿)
123 eqid 2795 . . 3 (0g𝐿) = (0g𝐿)
124 eqid 2795 . . 3 ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)}))
125122, 123, 124isdrng2 19202 . 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 396   = wceq 1522  wcel 2081  wne 2984  wral 3105  Vcvv 3437  cdif 3856  wss 3859  c0 4211  {csn 4472   cint 4782   ciun 4825   ciin 4826  cmpt 5041  dom cdm 5443  ran crn 5444  cfv 6225  (class class class)co 7016  Basecbs 16312  s cress 16313  0gc0g 16542  Grpcgrp 17861  SubGrpcsubg 18027  mulGrpcmgp 18929  Ringcrg 18987  DivRingcdr 19192  SubRingcsubrg 19221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-0g 16544  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-grp 17864  df-minusg 17865  df-subg 18030  df-mgp 18930  df-ur 18942  df-ring 18989  df-oppr 19063  df-dvdsr 19081  df-unit 19082  df-invr 19112  df-dvr 19123  df-drng 19194  df-subrg 19223
This theorem is referenced by:  sdrgint  19273  primefld  19274
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