| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | imasabl.u | . . . 4
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | 
| 2 |  | imasabl.v | . . . 4
⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | 
| 3 |  | imasabl.p | . . . 4
⊢ (𝜑 → + =
(+g‘𝑅)) | 
| 4 |  | imasabl.f | . . . 4
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) | 
| 5 |  | imasabl.e | . . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) | 
| 6 |  | imasabl.r | . . . . 5
⊢ (𝜑 → 𝑅 ∈ Abel) | 
| 7 | 6 | ablgrpd 19805 | . . . 4
⊢ (𝜑 → 𝑅 ∈ Grp) | 
| 8 |  | imasabl.z | . . . 4
⊢  0 =
(0g‘𝑅) | 
| 9 | 1, 2, 3, 4, 5, 7, 8 | imasgrp 19075 | . . 3
⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈))) | 
| 10 | 1, 2, 4, 6 | imasbas 17558 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐵 = (Base‘𝑈)) | 
| 11 | 10 | eqcomd 2742 | . . . . . . . . . 10
⊢ (𝜑 → (Base‘𝑈) = 𝐵) | 
| 12 | 11 | eleq2d 2826 | . . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (Base‘𝑈) ↔ 𝑥 ∈ 𝐵)) | 
| 13 | 11 | eleq2d 2826 | . . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ (Base‘𝑈) ↔ 𝑦 ∈ 𝐵)) | 
| 14 | 12, 13 | anbi12d 632 | . . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) | 
| 15 | 14 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ ((𝑥 ∈
(Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) | 
| 16 |  | foelcdmi 6969 | . . . . . . . . . . . 12
⊢ ((𝐹:𝑉–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥) | 
| 17 | 16 | ex 412 | . . . . . . . . . . 11
⊢ (𝐹:𝑉–onto→𝐵 → (𝑥 ∈ 𝐵 → ∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥)) | 
| 18 |  | foelcdmi 6969 | . . . . . . . . . . . 12
⊢ ((𝐹:𝑉–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦) | 
| 19 | 18 | ex 412 | . . . . . . . . . . 11
⊢ (𝐹:𝑉–onto→𝐵 → (𝑦 ∈ 𝐵 → ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦)) | 
| 20 | 17, 19 | anim12d 609 | . . . . . . . . . 10
⊢ (𝐹:𝑉–onto→𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦))) | 
| 21 | 4, 20 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦))) | 
| 22 | 21 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦))) | 
| 23 | 6 | ad3antrrr 730 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑅 ∈ Abel) | 
| 24 | 2 | eleq2d 2826 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑎 ∈ 𝑉 ↔ 𝑎 ∈ (Base‘𝑅))) | 
| 25 | 24 | biimpd 229 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑎 ∈ 𝑉 → 𝑎 ∈ (Base‘𝑅))) | 
| 26 | 25 | adantr 480 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ (𝑎 ∈ 𝑉 → 𝑎 ∈ (Base‘𝑅))) | 
| 27 | 26 | imp 406 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ (Base‘𝑅)) | 
| 28 | 27 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ (Base‘𝑅)) | 
| 29 | 2 | eleq2d 2826 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑏 ∈ 𝑉 ↔ 𝑏 ∈ (Base‘𝑅))) | 
| 30 | 29 | biimpd 229 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅))) | 
| 31 | 30 | adantr 480 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅))) | 
| 32 | 31 | adantr 480 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) → (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅))) | 
| 33 | 32 | imp 406 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ (Base‘𝑅)) | 
| 34 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . 19
⊢
(Base‘𝑅) =
(Base‘𝑅) | 
| 35 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . 19
⊢
(+g‘𝑅) = (+g‘𝑅) | 
| 36 | 34, 35 | ablcom 19818 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ Abel ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)𝑏) = (𝑏(+g‘𝑅)𝑎)) | 
| 37 | 23, 28, 33, 36 | syl3anc 1372 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → (𝑎(+g‘𝑅)𝑏) = (𝑏(+g‘𝑅)𝑎)) | 
| 38 | 37 | fveq2d 6909 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑏(+g‘𝑅)𝑎))) | 
| 39 |  | simplll 774 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝜑) | 
| 40 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | 
| 41 | 40 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉) | 
| 42 |  | simpr 484 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉) | 
| 43 | 3 | eqcomd 2742 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (+g‘𝑅) = + ) | 
| 44 | 43 | oveqd 7449 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑎(+g‘𝑅)𝑏) = (𝑎 + 𝑏)) | 
| 45 | 44 | fveq2d 6909 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑎 + 𝑏))) | 
| 46 | 43 | oveqd 7449 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑝(+g‘𝑅)𝑞) = (𝑝 + 𝑞)) | 
| 47 | 46 | fveq2d 6909 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐹‘(𝑝(+g‘𝑅)𝑞)) = (𝐹‘(𝑝 + 𝑞))) | 
| 48 | 45, 47 | eqeq12d 2752 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑝(+g‘𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) | 
| 49 | 48 | 3ad2ant1 1133 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑝(+g‘𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) | 
| 50 | 5, 49 | sylibrd 259 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑝(+g‘𝑅)𝑞)))) | 
| 51 |  | eqid 2736 | . . . . . . . . . . . . . . . . . 18
⊢
(+g‘𝑈) = (+g‘𝑈) | 
| 52 | 4, 50, 1, 2, 6, 35,
51 | imasaddval 17578 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏))) | 
| 53 | 39, 41, 42, 52 | syl3anc 1372 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏))) | 
| 54 | 4, 50, 1, 2, 6, 35,
51 | imasaddval 17578 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝐹‘(𝑏(+g‘𝑅)𝑎))) | 
| 55 | 39, 42, 41, 54 | syl3anc 1372 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝐹‘(𝑏(+g‘𝑅)𝑎))) | 
| 56 | 38, 53, 55 | 3eqtr4d 2786 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎))) | 
| 57 | 56 | adantr 480 | . . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎))) | 
| 58 |  | oveq12 7441 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐹‘𝑎) = 𝑥 ∧ (𝐹‘𝑏) = 𝑦) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝑥(+g‘𝑈)𝑦)) | 
| 59 | 58 | ancoms 458 | . . . . . . . . . . . . . . . 16
⊢ (((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝑥(+g‘𝑈)𝑦)) | 
| 60 |  | oveq12 7441 | . . . . . . . . . . . . . . . 16
⊢ (((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝑦(+g‘𝑈)𝑥)) | 
| 61 | 59, 60 | eqeq12d 2752 | . . . . . . . . . . . . . . 15
⊢ (((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥) → (((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) ↔ (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))) | 
| 62 | 61 | adantl 481 | . . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥)) → (((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) ↔ (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))) | 
| 63 | 57, 62 | mpbid 232 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥)) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)) | 
| 64 | 63 | exp32 420 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑏) = 𝑦 → ((𝐹‘𝑎) = 𝑥 → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))) | 
| 65 | 64 | rexlimdva 3154 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦 → ((𝐹‘𝑎) = 𝑥 → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))) | 
| 66 | 65 | com23 86 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ 𝑎 ∈ 𝑉) → ((𝐹‘𝑎) = 𝑥 → (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦 → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))) | 
| 67 | 66 | rexlimdva 3154 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ (∃𝑎 ∈
𝑉 (𝐹‘𝑎) = 𝑥 → (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦 → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))) | 
| 68 | 67 | impd 410 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ ((∃𝑎 ∈
𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))) | 
| 69 | 22, 68 | syld 47 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))) | 
| 70 | 15, 69 | sylbid 240 | . . . . . 6
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ ((𝑥 ∈
(Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))) | 
| 71 | 70 | imp 406 | . . . . 5
⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
∧ (𝑥 ∈
(Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)) | 
| 72 | 71 | ralrimivva 3201 | . . . 4
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ ∀𝑥 ∈
(Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)) | 
| 73 |  | simpr 484 | . . . 4
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ (𝑈 ∈ Grp ∧
(𝐹‘ 0 ) =
(0g‘𝑈))) | 
| 74 | 72, 73 | jca 511 | . . 3
⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))
→ (∀𝑥 ∈
(Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))) | 
| 75 | 9, 74 | mpdan 687 | . 2
⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))) | 
| 76 |  | eqid 2736 | . . . . 5
⊢
(Base‘𝑈) =
(Base‘𝑈) | 
| 77 | 76, 51 | isabl2 19809 | . . . 4
⊢ (𝑈 ∈ Abel ↔ (𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))) | 
| 78 | 77 | anbi1i 624 | . . 3
⊢ ((𝑈 ∈ Abel ∧ (𝐹‘ 0 ) =
(0g‘𝑈))
↔ ((𝑈 ∈ Grp ∧
∀𝑥 ∈
(Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)) ∧ (𝐹‘ 0 ) =
(0g‘𝑈))) | 
| 79 |  | an21 644 | . . 3
⊢ (((𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)) ∧ (𝐹‘ 0 ) =
(0g‘𝑈))
↔ (∀𝑥 ∈
(Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))) | 
| 80 | 78, 79 | bitri 275 | . 2
⊢ ((𝑈 ∈ Abel ∧ (𝐹‘ 0 ) =
(0g‘𝑈))
↔ (∀𝑥 ∈
(Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) =
(0g‘𝑈)))) | 
| 81 | 75, 80 | sylibr 234 | 1
⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) =
(0g‘𝑈))) |