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Theorem imasabl 19909
Description: The image structure of an abelian group is an abelian group (imasgrp 19087 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 19819 . . . 4 (𝜑𝑅 ∈ Grp)
8 imasabl.z . . . 4 0 = (0g𝑅)
91, 2, 3, 4, 5, 7, 8imasgrp 19087 . . 3 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
101, 2, 4, 6imasbas 17559 . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝑈))
1110eqcomd 2741 . . . . . . . . . 10 (𝜑 → (Base‘𝑈) = 𝐵)
1211eleq2d 2825 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝑈) ↔ 𝑥𝐵))
1311eleq2d 2825 . . . . . . . . 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 6970 . . . . . . . . . . . 12 ((𝐹:𝑉onto𝐵𝑥𝐵) → ∃𝑎𝑉 (𝐹𝑎) = 𝑥)
1716ex 412 . . . . . . . . . . 11 (𝐹:𝑉onto𝐵 → (𝑥𝐵 → ∃𝑎𝑉 (𝐹𝑎) = 𝑥))
18 foelcdmi 6970 . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) = (Base‘𝑅)
35 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (+g𝑅) = (+g𝑅)
3634, 35ablcom 19832 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Abel ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g𝑅)𝑏) = (𝑏(+g𝑅)𝑎))
3723, 28, 33, 36syl3anc 1370 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → (𝑎(+g𝑅)𝑏) = (𝑏(+g𝑅)𝑎))
3837fveq2d 6911 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
39 simplll 775 . . . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (+g𝑅) = + )
4443oveqd 7448 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎(+g𝑅)𝑏) = (𝑎 + 𝑏))
4544fveq2d 6911 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑎 + 𝑏)))
4643oveqd 7448 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑝(+g𝑅)𝑞) = (𝑝 + 𝑞))
4746fveq2d 6911 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐹‘(𝑝(+g𝑅)𝑞)) = (𝐹‘(𝑝 + 𝑞)))
4845, 47eqeq12d 2751 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
49483ad2ant1 1132 . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . 18 (+g𝑈) = (+g𝑈)
524, 50, 1, 2, 6, 35, 51imasaddval 17579 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑉𝑏𝑉) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
5339, 41, 42, 52syl3anc 1370 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
544, 50, 1, 2, 6, 35, 51imasaddval 17579 . . . . . . . . . . . . . . . . 17 ((𝜑𝑏𝑉𝑎𝑉) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
5539, 42, 41, 54syl3anc 1370 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
5638, 53, 553eqtr4d 2785 . . . . . . . . . . . . . . 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 7440 . . . . . . . . . . . . . . . . 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 7440 . . . . . . . . . . . . . . . 16 (((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝑦(+g𝑈)𝑥))
6159, 60eqeq12d 2751 . . . . . . . . . . . . . . 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 3153 . . . . . . . . . . 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 3153 . . . . . . . . 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 3200 . . . 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 2735 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
7776, 51isabl2 19823 . . . 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 1537  wcel 2106  wral 3059  wrex 3068  ontowfo 6561  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  s cimas 17551  Grpcgrp 18964  Abelcabl 19814
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-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  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-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  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-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  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 17488  df-imas 17555  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-cmn 19815  df-abl 19816
This theorem is referenced by:  imasrng  20195
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