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Theorem imasabl 19895
Description: The image structure of an abelian group is an abelian group (imasgrp 19075 analog). (Contributed by AV, 22-Feb-2025.)
Hypotheses
Ref Expression
imasabl.u (𝜑𝑈 = (𝐹s 𝑅))
imasabl.v (𝜑𝑉 = (Base‘𝑅))
imasabl.p (𝜑+ = (+g𝑅))
imasabl.f (𝜑𝐹:𝑉onto𝐵)
imasabl.e ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasabl.r (𝜑𝑅 ∈ Abel)
imasabl.z 0 = (0g𝑅)
Assertion
Ref Expression
imasabl (𝜑 → (𝑈 ∈ Abel ∧ (𝐹0 ) = (0g𝑈)))
Distinct variable groups:   𝐵,𝑎,𝑏,𝑝,𝑞   𝐹,𝑎,𝑏,𝑝,𝑞   𝑅,𝑝,𝑞   𝑈,𝑎,𝑏,𝑝,𝑞   𝑉,𝑎,𝑏,𝑝,𝑞   + ,𝑝,𝑞   0 ,𝑎,𝑏,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞
Allowed substitution hints:   + (𝑎,𝑏)   𝑅(𝑎,𝑏)

Proof of Theorem imasabl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasabl.u . . . 4 (𝜑𝑈 = (𝐹s 𝑅))
2 imasabl.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
3 imasabl.p . . . 4 (𝜑+ = (+g𝑅))
4 imasabl.f . . . 4 (𝜑𝐹:𝑉onto𝐵)
5 imasabl.e . . . 4 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
6 imasabl.r . . . . 5 (𝜑𝑅 ∈ Abel)
76ablgrpd 19805 . . . 4 (𝜑𝑅 ∈ Grp)
8 imasabl.z . . . 4 0 = (0g𝑅)
91, 2, 3, 4, 5, 7, 8imasgrp 19075 . . 3 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
101, 2, 4, 6imasbas 17558 . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝑈))
1110eqcomd 2742 . . . . . . . . . 10 (𝜑 → (Base‘𝑈) = 𝐵)
1211eleq2d 2826 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝑈) ↔ 𝑥𝐵))
1311eleq2d 2826 . . . . . . . . 9 (𝜑 → (𝑦 ∈ (Base‘𝑈) ↔ 𝑦𝐵))
1412, 13anbi12d 632 . . . . . . . 8 (𝜑 → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥𝐵𝑦𝐵)))
1514adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥𝐵𝑦𝐵)))
16 foelcdmi 6969 . . . . . . . . . . . 12 ((𝐹:𝑉onto𝐵𝑥𝐵) → ∃𝑎𝑉 (𝐹𝑎) = 𝑥)
1716ex 412 . . . . . . . . . . 11 (𝐹:𝑉onto𝐵 → (𝑥𝐵 → ∃𝑎𝑉 (𝐹𝑎) = 𝑥))
18 foelcdmi 6969 . . . . . . . . . . . 12 ((𝐹:𝑉onto𝐵𝑦𝐵) → ∃𝑏𝑉 (𝐹𝑏) = 𝑦)
1918ex 412 . . . . . . . . . . 11 (𝐹:𝑉onto𝐵 → (𝑦𝐵 → ∃𝑏𝑉 (𝐹𝑏) = 𝑦))
2017, 19anim12d 609 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ((𝑥𝐵𝑦𝐵) → (∃𝑎𝑉 (𝐹𝑎) = 𝑥 ∧ ∃𝑏𝑉 (𝐹𝑏) = 𝑦)))
214, 20syl 17 . . . . . . . . 9 (𝜑 → ((𝑥𝐵𝑦𝐵) → (∃𝑎𝑉 (𝐹𝑎) = 𝑥 ∧ ∃𝑏𝑉 (𝐹𝑏) = 𝑦)))
2221adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ((𝑥𝐵𝑦𝐵) → (∃𝑎𝑉 (𝐹𝑎) = 𝑥 ∧ ∃𝑏𝑉 (𝐹𝑏) = 𝑦)))
236ad3antrrr 730 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑅 ∈ Abel)
242eleq2d 2826 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎𝑉𝑎 ∈ (Base‘𝑅)))
2524biimpd 229 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑎𝑉𝑎 ∈ (Base‘𝑅)))
2625adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (𝑎𝑉𝑎 ∈ (Base‘𝑅)))
2726imp 406 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) → 𝑎 ∈ (Base‘𝑅))
2827adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑎 ∈ (Base‘𝑅))
292eleq2d 2826 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑏𝑉𝑏 ∈ (Base‘𝑅)))
3029biimpd 229 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑏𝑉𝑏 ∈ (Base‘𝑅)))
3130adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (𝑏𝑉𝑏 ∈ (Base‘𝑅)))
3231adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) → (𝑏𝑉𝑏 ∈ (Base‘𝑅)))
3332imp 406 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑏 ∈ (Base‘𝑅))
34 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) = (Base‘𝑅)
35 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (+g𝑅) = (+g𝑅)
3634, 35ablcom 19818 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Abel ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g𝑅)𝑏) = (𝑏(+g𝑅)𝑎))
3723, 28, 33, 36syl3anc 1372 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → (𝑎(+g𝑅)𝑏) = (𝑏(+g𝑅)𝑎))
3837fveq2d 6909 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
39 simplll 774 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝜑)
40 simpr 484 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) → 𝑎𝑉)
4140adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑎𝑉)
42 simpr 484 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑏𝑉)
433eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (+g𝑅) = + )
4443oveqd 7449 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎(+g𝑅)𝑏) = (𝑎 + 𝑏))
4544fveq2d 6909 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑎 + 𝑏)))
4643oveqd 7449 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑝(+g𝑅)𝑞) = (𝑝 + 𝑞))
4746fveq2d 6909 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐹‘(𝑝(+g𝑅)𝑞)) = (𝐹‘(𝑝 + 𝑞)))
4845, 47eqeq12d 2752 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
49483ad2ant1 1133 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
505, 49sylibrd 259 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
51 eqid 2736 . . . . . . . . . . . . . . . . . 18 (+g𝑈) = (+g𝑈)
524, 50, 1, 2, 6, 35, 51imasaddval 17578 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑉𝑏𝑉) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
5339, 41, 42, 52syl3anc 1372 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
544, 50, 1, 2, 6, 35, 51imasaddval 17578 . . . . . . . . . . . . . . . . 17 ((𝜑𝑏𝑉𝑎𝑉) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
5539, 42, 41, 54syl3anc 1372 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
5638, 53, 553eqtr4d 2786 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = ((𝐹𝑏)(+g𝑈)(𝐹𝑎)))
5756adantr 480 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) ∧ ((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥)) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = ((𝐹𝑏)(+g𝑈)(𝐹𝑎)))
58 oveq12 7441 . . . . . . . . . . . . . . . . 17 (((𝐹𝑎) = 𝑥 ∧ (𝐹𝑏) = 𝑦) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝑥(+g𝑈)𝑦))
5958ancoms 458 . . . . . . . . . . . . . . . 16 (((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝑥(+g𝑈)𝑦))
60 oveq12 7441 . . . . . . . . . . . . . . . 16 (((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝑦(+g𝑈)𝑥))
6159, 60eqeq12d 2752 . . . . . . . . . . . . . . 15 (((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥) → (((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) ↔ (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
6261adantl 481 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) ∧ ((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥)) → (((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) ↔ (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
6357, 62mpbid 232 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) ∧ ((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥)) → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))
6463exp32 420 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑏) = 𝑦 → ((𝐹𝑎) = 𝑥 → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))))
6564rexlimdva 3154 . . . . . . . . . . 11 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) → (∃𝑏𝑉 (𝐹𝑏) = 𝑦 → ((𝐹𝑎) = 𝑥 → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))))
6665com23 86 . . . . . . . . . 10 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) → ((𝐹𝑎) = 𝑥 → (∃𝑏𝑉 (𝐹𝑏) = 𝑦 → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))))
6766rexlimdva 3154 . . . . . . . . 9 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (∃𝑎𝑉 (𝐹𝑎) = 𝑥 → (∃𝑏𝑉 (𝐹𝑏) = 𝑦 → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))))
6867impd 410 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ((∃𝑎𝑉 (𝐹𝑎) = 𝑥 ∧ ∃𝑏𝑉 (𝐹𝑏) = 𝑦) → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
6922, 68syld 47 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ((𝑥𝐵𝑦𝐵) → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
7015, 69sylbid 240 . . . . . 6 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
7170imp 406 . . . . 5 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))
7271ralrimivva 3201 . . . 4 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))
73 simpr 484 . . . 4 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
7472, 73jca 511 . . 3 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))))
759, 74mpdan 687 . 2 (𝜑 → (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))))
76 eqid 2736 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
7776, 51isabl2 19809 . . . 4 (𝑈 ∈ Abel ↔ (𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
7877anbi1i 624 . . 3 ((𝑈 ∈ Abel ∧ (𝐹0 ) = (0g𝑈)) ↔ ((𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)) ∧ (𝐹0 ) = (0g𝑈)))
79 an21 644 . . 3 (((𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)) ∧ (𝐹0 ) = (0g𝑈)) ↔ (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))))
8078, 79bitri 275 . 2 ((𝑈 ∈ Abel ∧ (𝐹0 ) = (0g𝑈)) ↔ (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))))
8175, 80sylibr 234 1 (𝜑 → (𝑈 ∈ Abel ∧ (𝐹0 ) = (0g𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  ontowfo 6558  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  0gc0g 17485  s cimas 17550  Grpcgrp 18952  Abelcabl 19800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17487  df-imas 17554  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-cmn 19801  df-abl 19802
This theorem is referenced by:  imasrng  20175
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