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Theorem subdrgint 19578
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 19553 . . . 4 ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubRing‘𝑅))
41, 2, 3syl2anc 587 . . 3 (𝜑 𝑆 ∈ (SubRing‘𝑅))
5 subdrgint.1 . . . 4 𝐿 = (𝑅s 𝑆)
65subrgring 19534 . . 3 ( 𝑆 ∈ (SubRing‘𝑅) → 𝐿 ∈ Ring)
74, 6syl 17 . 2 (𝜑𝐿 ∈ Ring)
85fveq2i 6652 . . . 4 (mulGrp‘𝐿) = (mulGrp‘(𝑅s 𝑆))
98oveq1i 7149 . . 3 ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)}))
10 subdrgint.2 . . . . . . 7 (𝜑𝑅 ∈ DivRing)
11 eqid 2801 . . . . . . . 8 (𝑅s 𝑆) = (𝑅s 𝑆)
12 eqid 2801 . . . . . . . 8 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1311, 12mgpress 19246 . . . . . . 7 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → ((mulGrp‘𝑅) ↾s 𝑆) = (mulGrp‘(𝑅s 𝑆)))
1410, 4, 13syl2anc 587 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s 𝑆) = (mulGrp‘(𝑅s 𝑆)))
1514oveq1d 7154 . . . . 5 (𝜑 → (((mulGrp‘𝑅) ↾s 𝑆) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
16 difssd 4063 . . . . . . 7 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ (Base‘𝐿))
17 eqid 2801 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
1817subrgss 19532 . . . . . . . 8 ( 𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ (Base‘𝑅))
195, 17ressbas2 16550 . . . . . . . 8 ( 𝑆 ⊆ (Base‘𝑅) → 𝑆 = (Base‘𝐿))
204, 18, 193syl 18 . . . . . . 7 (𝜑 𝑆 = (Base‘𝐿))
2116, 20sseqtrrd 3959 . . . . . 6 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) ⊆ 𝑆)
22 ressabs 16558 . . . . . 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 587 . . . . 5 (𝜑 → (((mulGrp‘𝑅) ↾s 𝑆) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
2415, 23eqtr3d 2838 . . . 4 (𝜑 → ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})))
25 intiin 4949 . . . . . . . 8 𝑆 = 𝑠𝑆 𝑠
2625, 20syl5reqr 2851 . . . . . . 7 (𝜑 → (Base‘𝐿) = 𝑠𝑆 𝑠)
2726difeq1d 4052 . . . . . 6 (𝜑 → ((Base‘𝐿) ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
2827oveq2d 7155 . . . . 5 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})))
29 vex 3447 . . . . . . . . . 10 𝑠 ∈ V
3029difexi 5199 . . . . . . . . 9 (𝑠 ∖ {(0g𝐿)}) ∈ V
3130dfiin3 5807 . . . . . . . 8 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))
32 iindif1 4963 . . . . . . . . 9 (𝑆 ≠ ∅ → 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
332, 32syl 17 . . . . . . . 8 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
3431, 33syl5eqr 2850 . . . . . . 7 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)}))
3534oveq2d 7155 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})))
36 difss 4062 . . . . . . . . . 10 ((Base‘𝑅) ∖ {(0g𝑅)}) ⊆ (Base‘𝑅)
37 eqid 2801 . . . . . . . . . . 11 ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))
3812, 17mgpbas 19241 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
3937, 38ressbas2 16550 . . . . . . . . . 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 6663 . . . . . . . 8 ((Base‘𝑅) ∖ {(0g𝑅)}) ∈ V
42 iinssiun 4897 . . . . . . . . . . 11 (𝑆 ≠ ∅ → 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}))
432, 42syl 17 . . . . . . . . . 10 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}))
44 subrgsubg 19537 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ∈ (SubGrp‘𝑅))
4544ssriv 3922 . . . . . . . . . . . . . . . . . 18 (SubRing‘𝑅) ⊆ (SubGrp‘𝑅)
461, 45sstrdi 3930 . . . . . . . . . . . . . . . . 17 (𝜑𝑆 ⊆ (SubGrp‘𝑅))
47 subgint 18298 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ (SubGrp‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubGrp‘𝑅))
4846, 2, 47syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑 𝑆 ∈ (SubGrp‘𝑅))
49 eqid 2801 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
505, 49subg0 18280 . . . . . . . . . . . . . . . 16 ( 𝑆 ∈ (SubGrp‘𝑅) → (0g𝑅) = (0g𝐿))
5148, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0g𝑅) = (0g𝐿))
5251adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → (0g𝑅) = (0g𝐿))
5352sneqd 4540 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → {(0g𝑅)} = {(0g𝐿)})
5453difeq2d 4053 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) = (𝑠 ∖ {(0g𝐿)}))
551sselda 3918 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → 𝑠 ∈ (SubRing‘𝑅))
5617subrgss 19532 . . . . . . . . . . . . . 14 (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ⊆ (Base‘𝑅))
5755, 56syl 17 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → 𝑠 ⊆ (Base‘𝑅))
5857ssdifd 4071 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
5954, 58eqsstrrd 3957 . . . . . . . . . . 11 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6059iunssd 4940 . . . . . . . . . 10 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6143, 60sstrd 3928 . . . . . . . . 9 (𝜑 𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6231, 61eqsstrrid 3967 . . . . . . . 8 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
63 ressabs 16558 . . . . . . . 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 590 . . . . . . 7 (𝜑 → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))))
6517, 49, 37drngmgp 19510 . . . . . . . . . . . . . 14 (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6610, 65syl 17 . . . . . . . . . . . . 13 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6766adantr 484 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ∈ Grp)
6859, 40sseqtrdi 3968 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
69 ressabs 16558 . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
71 eqid 2801 . . . . . . . . . . . . . . . . . 18 (𝑅s 𝑠) = (𝑅s 𝑠)
7271, 12mgpress 19246 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ DivRing ∧ 𝑠𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅s 𝑠)))
7310, 72sylan 583 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅s 𝑠)))
7454eqcomd 2807 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) = (𝑠 ∖ {(0g𝑅)}))
7573, 74oveq12d 7157 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})))
76 simpr 488 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → 𝑠𝑆)
77 difssd 4063 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠)
78 ressabs 16558 . . . . . . . . . . . . . . . 16 ((𝑠𝑆 ∧ (𝑠 ∖ {(0g𝐿)}) ⊆ 𝑠) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
7976, 77, 78syl2anc 587 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
8075, 79eqtr3d 2838 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})))
8171, 17ressbas2 16550 . . . . . . . . . . . . . . . . . 18 (𝑠 ⊆ (Base‘𝑅) → 𝑠 = (Base‘(𝑅s 𝑠)))
8255, 56, 813syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠𝑆) → 𝑠 = (Base‘(𝑅s 𝑠)))
8371, 49subrg0 19538 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (SubRing‘𝑅) → (0g𝑅) = (0g‘(𝑅s 𝑠)))
8455, 83syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠𝑆) → (0g𝑅) = (0g‘(𝑅s 𝑠)))
8584sneqd 4540 . . . . . . . . . . . . . . . . 17 ((𝜑𝑠𝑆) → {(0g𝑅)} = {(0g‘(𝑅s 𝑠))})
8682, 85difeq12d 4054 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (𝑠 ∖ {(0g𝑅)}) = ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))}))
8786oveq2d 7155 . . . . . . . . . . . . . . 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 2801 . . . . . . . . . . . . . . . . 17 (Base‘(𝑅s 𝑠)) = (Base‘(𝑅s 𝑠))
90 eqid 2801 . . . . . . . . . . . . . . . . 17 (0g‘(𝑅s 𝑠)) = (0g‘(𝑅s 𝑠))
91 eqid 2801 . . . . . . . . . . . . . . . . 17 ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))})) = ((mulGrp‘(𝑅s 𝑠)) ↾s ((Base‘(𝑅s 𝑠)) ∖ {(0g‘(𝑅s 𝑠))}))
9289, 90, 91drngmgp 19510 . . . . . . . . . . . . . . . 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 2893 . . . . . . . . . . . . . 14 ((𝜑𝑠𝑆) → ((mulGrp‘(𝑅s 𝑠)) ↾s (𝑠 ∖ {(0g𝑅)})) ∈ Grp)
9580, 94eqeltrrd 2894 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp)
9670, 95eqeltrd 2893 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s (𝑠 ∖ {(0g𝐿)})) ∈ Grp)
97 eqid 2801 . . . . . . . . . . . . 13 (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})))
9897issubg 18274 . . . . . . . . . . . 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 3152 . . . . . . . . . 10 (𝜑 → ∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
101 eqid 2801 . . . . . . . . . . 11 (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))
102101rnmptss 6867 . . . . . . . . . 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 6067 . . . . . . . . . . . . 13 (∀𝑠𝑆 (𝑠 ∖ {(0g𝐿)}) ∈ V → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆)
105 difexg 5198 . . . . . . . . . . . . 13 (𝑠𝑆 → (𝑠 ∖ {(0g𝐿)}) ∈ V)
106104, 105mprg 3123 . . . . . . . . . . . 12 dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆
107106a1i 11 . . . . . . . . . . 11 (𝜑 → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = 𝑆)
108107, 2eqnetrd 3057 . . . . . . . . . 10 (𝜑 → dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
109 dm0rn0 5763 . . . . . . . . . . 11 (dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ∅ ↔ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) = ∅)
110109necon3bii 3042 . . . . . . . . . 10 (dom (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅ ↔ ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
111108, 110sylib 221 . . . . . . . . 9 (𝜑 → ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ≠ ∅)
112 subgint 18298 . . . . . . . . 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 587 . . . . . . . 8 (𝜑 ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)}))))
114 eqid 2801 . . . . . . . . 9 (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) = (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g𝑅)})) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)})))
115114subggrp 18277 . . . . . . . 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 2894 . . . . . 6 (𝜑 → ((mulGrp‘𝑅) ↾s ran (𝑠𝑆 ↦ (𝑠 ∖ {(0g𝐿)}))) ∈ Grp)
11835, 117eqeltrrd 2894 . . . . 5 (𝜑 → ((mulGrp‘𝑅) ↾s ( 𝑠𝑆 𝑠 ∖ {(0g𝐿)})) ∈ Grp)
11928, 118eqeltrd 2893 . . . 4 (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
12024, 119eqeltrd 2893 . . 3 (𝜑 → ((mulGrp‘(𝑅s 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
1219, 120eqeltrid 2897 . 2 (𝜑 → ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp)
122 eqid 2801 . . 3 (Base‘𝐿) = (Base‘𝐿)
123 eqid 2801 . . 3 (0g𝐿) = (0g𝐿)
124 eqid 2801 . . 3 ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) = ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)}))
125122, 123, 124isdrng2 19508 . 2 (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g𝐿)})) ∈ Grp))
1267, 121, 125sylanbrc 586 1 (𝜑𝐿 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  Vcvv 3444  cdif 3881  wss 3884  c0 4246  {csn 4528   cint 4841   ciun 4884   ciin 4885  cmpt 5113  dom cdm 5523  ran crn 5524  cfv 6328  (class class class)co 7139  Basecbs 16478  s cress 16479  0gc0g 16708  Grpcgrp 18098  SubGrpcsubg 18268  mulGrpcmgp 19235  Ringcrg 19293  DivRingcdr 19498  SubRingcsubrg 19527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-minusg 18102  df-subg 18271  df-mgp 19236  df-ur 19248  df-ring 19295  df-oppr 19372  df-dvdsr 19390  df-unit 19391  df-invr 19421  df-dvr 19432  df-drng 19500  df-subrg 19529
This theorem is referenced by:  sdrgint  19579  primefld  19580
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