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Theorem imasabl 19838
Description: The image structure of an abelian group is an abelian group (imasgrp 19019 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 19748 . . . 4 (𝜑𝑅 ∈ Grp)
8 imasabl.z . . . 4 0 = (0g𝑅)
91, 2, 3, 4, 5, 7, 8imasgrp 19019 . . 3 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
101, 2, 4, 6imasbas 17501 . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝑈))
1110eqcomd 2734 . . . . . . . . . 10 (𝜑 → (Base‘𝑈) = 𝐵)
1211eleq2d 2815 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝑈) ↔ 𝑥𝐵))
1311eleq2d 2815 . . . . . . . . 9 (𝜑 → (𝑦 ∈ (Base‘𝑈) ↔ 𝑦𝐵))
1412, 13anbi12d 630 . . . . . . . 8 (𝜑 → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥𝐵𝑦𝐵)))
1514adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥𝐵𝑦𝐵)))
16 foelcdmi 6965 . . . . . . . . . . . 12 ((𝐹:𝑉onto𝐵𝑥𝐵) → ∃𝑎𝑉 (𝐹𝑎) = 𝑥)
1716ex 411 . . . . . . . . . . 11 (𝐹:𝑉onto𝐵 → (𝑥𝐵 → ∃𝑎𝑉 (𝐹𝑎) = 𝑥))
18 foelcdmi 6965 . . . . . . . . . . . 12 ((𝐹:𝑉onto𝐵𝑦𝐵) → ∃𝑏𝑉 (𝐹𝑏) = 𝑦)
1918ex 411 . . . . . . . . . . 11 (𝐹:𝑉onto𝐵 → (𝑦𝐵 → ∃𝑏𝑉 (𝐹𝑏) = 𝑦))
2017, 19anim12d 607 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ((𝑥𝐵𝑦𝐵) → (∃𝑎𝑉 (𝐹𝑎) = 𝑥 ∧ ∃𝑏𝑉 (𝐹𝑏) = 𝑦)))
214, 20syl 17 . . . . . . . . 9 (𝜑 → ((𝑥𝐵𝑦𝐵) → (∃𝑎𝑉 (𝐹𝑎) = 𝑥 ∧ ∃𝑏𝑉 (𝐹𝑏) = 𝑦)))
2221adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ((𝑥𝐵𝑦𝐵) → (∃𝑎𝑉 (𝐹𝑎) = 𝑥 ∧ ∃𝑏𝑉 (𝐹𝑏) = 𝑦)))
236ad3antrrr 728 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑅 ∈ Abel)
242eleq2d 2815 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎𝑉𝑎 ∈ (Base‘𝑅)))
2524biimpd 228 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑎𝑉𝑎 ∈ (Base‘𝑅)))
2625adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (𝑎𝑉𝑎 ∈ (Base‘𝑅)))
2726imp 405 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) → 𝑎 ∈ (Base‘𝑅))
2827adantr 479 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑎 ∈ (Base‘𝑅))
292eleq2d 2815 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑏𝑉𝑏 ∈ (Base‘𝑅)))
3029biimpd 228 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑏𝑉𝑏 ∈ (Base‘𝑅)))
3130adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (𝑏𝑉𝑏 ∈ (Base‘𝑅)))
3231adantr 479 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) → (𝑏𝑉𝑏 ∈ (Base‘𝑅)))
3332imp 405 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑏 ∈ (Base‘𝑅))
34 eqid 2728 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) = (Base‘𝑅)
35 eqid 2728 . . . . . . . . . . . . . . . . . . 19 (+g𝑅) = (+g𝑅)
3634, 35ablcom 19761 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Abel ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g𝑅)𝑏) = (𝑏(+g𝑅)𝑎))
3723, 28, 33, 36syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → (𝑎(+g𝑅)𝑏) = (𝑏(+g𝑅)𝑎))
3837fveq2d 6906 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
39 simplll 773 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝜑)
40 simpr 483 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) → 𝑎𝑉)
4140adantr 479 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑎𝑉)
42 simpr 483 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → 𝑏𝑉)
433eqcomd 2734 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (+g𝑅) = + )
4443oveqd 7443 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎(+g𝑅)𝑏) = (𝑎 + 𝑏))
4544fveq2d 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑎 + 𝑏)))
4643oveqd 7443 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑝(+g𝑅)𝑞) = (𝑝 + 𝑞))
4746fveq2d 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐹‘(𝑝(+g𝑅)𝑞)) = (𝐹‘(𝑝 + 𝑞)))
4845, 47eqeq12d 2744 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
49483ad2ant1 1130 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
505, 49sylibrd 258 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
51 eqid 2728 . . . . . . . . . . . . . . . . . 18 (+g𝑈) = (+g𝑈)
524, 50, 1, 2, 6, 35, 51imasaddval 17521 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑉𝑏𝑉) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
5339, 41, 42, 52syl3anc 1368 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
544, 50, 1, 2, 6, 35, 51imasaddval 17521 . . . . . . . . . . . . . . . . 17 ((𝜑𝑏𝑉𝑎𝑉) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
5539, 42, 41, 54syl3anc 1368 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝐹‘(𝑏(+g𝑅)𝑎)))
5638, 53, 553eqtr4d 2778 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = ((𝐹𝑏)(+g𝑈)(𝐹𝑎)))
5756adantr 479 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) ∧ ((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥)) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = ((𝐹𝑏)(+g𝑈)(𝐹𝑎)))
58 oveq12 7435 . . . . . . . . . . . . . . . . 17 (((𝐹𝑎) = 𝑥 ∧ (𝐹𝑏) = 𝑦) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝑥(+g𝑈)𝑦))
5958ancoms 457 . . . . . . . . . . . . . . . 16 (((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥) → ((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = (𝑥(+g𝑈)𝑦))
60 oveq12 7435 . . . . . . . . . . . . . . . 16 (((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥) → ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) = (𝑦(+g𝑈)𝑥))
6159, 60eqeq12d 2744 . . . . . . . . . . . . . . 15 (((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥) → (((𝐹𝑎)(+g𝑈)(𝐹𝑏)) = ((𝐹𝑏)(+g𝑈)(𝐹𝑎)) ↔ (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
6261adantl 480 . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) ∧ ((𝐹𝑏) = 𝑦 ∧ (𝐹𝑎) = 𝑥)) → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))
6463exp32 419 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ 𝑎𝑉) ∧ 𝑏𝑉) → ((𝐹𝑏) = 𝑦 → ((𝐹𝑎) = 𝑥 → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))))
6564rexlimdva 3152 . . . . . . . . . . 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 3152 . . . . . . . . 9 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (∃𝑎𝑉 (𝐹𝑎) = 𝑥 → (∃𝑏𝑉 (𝐹𝑏) = 𝑦 → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))))
6867impd 409 . . . . . . . 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 239 . . . . . 6 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
7170imp 405 . . . . 5 (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))
7271ralrimivva 3198 . . . 4 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥))
73 simpr 483 . . . 4 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
7472, 73jca 510 . . 3 ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))) → (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))))
759, 74mpdan 685 . 2 (𝜑 → (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))))
76 eqid 2728 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
7776, 51isabl2 19752 . . . 4 (𝑈 ∈ Abel ↔ (𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)))
7877anbi1i 622 . . 3 ((𝑈 ∈ Abel ∧ (𝐹0 ) = (0g𝑈)) ↔ ((𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥)) ∧ (𝐹0 ) = (0g𝑈)))
79 an21 642 . . 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 274 . 2 ((𝑈 ∈ Abel ∧ (𝐹0 ) = (0g𝑈)) ↔ (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g𝑈)𝑦) = (𝑦(+g𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈))))
8175, 80sylibr 233 1 (𝜑 → (𝑈 ∈ Abel ∧ (𝐹0 ) = (0g𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  wrex 3067  ontowfo 6551  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  0gc0g 17428  s cimas 17493  Grpcgrp 18897  Abelcabl 19743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-inf 9474  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-tset 17259  df-ple 17260  df-ds 17262  df-0g 17430  df-imas 17497  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-cmn 19744  df-abl 19745
This theorem is referenced by:  imasrng  20124
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