| Step | Hyp | Ref
| Expression |
| 1 | | sylow3.g |
. . 3
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 2 | | ovex 7464 |
. . 3
⊢ (𝑃 pSyl 𝐺) ∈ V |
| 3 | 1, 2 | jctir 520 |
. 2
⊢ (𝜑 → (𝐺 ∈ Grp ∧ (𝑃 pSyl 𝐺) ∈ V)) |
| 4 | | sylow3.xf |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈ Fin) |
| 5 | | sylow3.p |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ∈ ℙ) |
| 6 | | sylow3.x |
. . . . . . . . . . . 12
⊢ 𝑋 = (Base‘𝐺) |
| 7 | 6 | fislw 19643 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝑦 ∈ (𝑃 pSyl 𝐺) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))) |
| 8 | 1, 4, 5, 7 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ (𝑃 pSyl 𝐺) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))) |
| 9 | 8 | biimpa 476 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺)) → (𝑦 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) |
| 10 | 9 | adantrl 716 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → (𝑦 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) |
| 11 | 10 | simpld 494 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑦 ∈ (SubGrp‘𝐺)) |
| 12 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑥 ∈ 𝑋) |
| 13 | | sylow3lem1.a |
. . . . . . . 8
⊢ + =
(+g‘𝐺) |
| 14 | | sylow3lem1.d |
. . . . . . . 8
⊢ − =
(-g‘𝐺) |
| 15 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) |
| 16 | 6, 13, 14, 15 | conjsubg 19268 |
. . . . . . 7
⊢ ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (SubGrp‘𝐺)) |
| 17 | 11, 12, 16 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (SubGrp‘𝐺)) |
| 18 | 6, 13, 14, 15 | conjsubgen 19269 |
. . . . . . . . 9
⊢ ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) |
| 19 | 11, 12, 18 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) |
| 20 | 4 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑋 ∈ Fin) |
| 21 | 6 | subgss 19145 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (SubGrp‘𝐺) → 𝑦 ⊆ 𝑋) |
| 22 | 11, 21 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑦 ⊆ 𝑋) |
| 23 | 20, 22 | ssfid 9301 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → 𝑦 ∈ Fin) |
| 24 | 6 | subgss 19145 |
. . . . . . . . . . 11
⊢ (ran
(𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (SubGrp‘𝐺) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ⊆ 𝑋) |
| 25 | 17, 24 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ⊆ 𝑋) |
| 26 | 20, 25 | ssfid 9301 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ Fin) |
| 27 | | hashen 14386 |
. . . . . . . . 9
⊢ ((𝑦 ∈ Fin ∧ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ Fin) → ((♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ↔ 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))) |
| 28 | 23, 26, 27 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → ((♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ↔ 𝑦 ≈ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))) |
| 29 | 19, 28 | mpbird 257 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → (♯‘𝑦) = (♯‘ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))) |
| 30 | 10 | simprd 495 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → (♯‘𝑦) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
| 31 | 29, 30 | eqtr3d 2779 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → (♯‘ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
| 32 | 6 | fislw 19643 |
. . . . . . . 8
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (ran
(𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (𝑃 pSyl 𝐺) ↔ (ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (SubGrp‘𝐺) ∧ (♯‘ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))) |
| 33 | 1, 4, 5, 32 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (𝑃 pSyl 𝐺) ↔ (ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (SubGrp‘𝐺) ∧ (♯‘ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))) |
| 34 | 33 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → (ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (𝑃 pSyl 𝐺) ↔ (ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (SubGrp‘𝐺) ∧ (♯‘ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))) |
| 35 | 17, 31, 34 | mpbir2and 713 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ (𝑃 pSyl 𝐺))) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (𝑃 pSyl 𝐺)) |
| 36 | 35 | ralrimivva 3202 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ (𝑃 pSyl 𝐺)ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (𝑃 pSyl 𝐺)) |
| 37 | | sylow3lem1.m |
. . . . 5
⊢ ⊕ =
(𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) |
| 38 | 37 | fmpo 8093 |
. . . 4
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ (𝑃 pSyl 𝐺)ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) ∈ (𝑃 pSyl 𝐺) ↔ ⊕ :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺)) |
| 39 | 36, 38 | sylib 218 |
. . 3
⊢ (𝜑 → ⊕ :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺)) |
| 40 | 1 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → 𝐺 ∈ Grp) |
| 41 | | eqid 2737 |
. . . . . . . . 9
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 42 | 6, 41 | grpidcl 18983 |
. . . . . . . 8
⊢ (𝐺 ∈ Grp →
(0g‘𝐺)
∈ 𝑋) |
| 43 | 40, 42 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → (0g‘𝐺) ∈ 𝑋) |
| 44 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → 𝑎 ∈ (𝑃 pSyl 𝐺)) |
| 45 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎) |
| 46 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → 𝑥 = (0g‘𝐺)) |
| 47 | 46 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((0g‘𝐺) + 𝑧)) |
| 48 | 47, 46 | oveq12d 7449 |
. . . . . . . . . 10
⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → ((𝑥 + 𝑧) − 𝑥) = (((0g‘𝐺) + 𝑧) −
(0g‘𝐺))) |
| 49 | 45, 48 | mpteq12dv 5233 |
. . . . . . . . 9
⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺)))) |
| 50 | 49 | rneqd 5949 |
. . . . . . . 8
⊢ ((𝑥 = (0g‘𝐺) ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺)))) |
| 51 | | vex 3484 |
. . . . . . . . . 10
⊢ 𝑎 ∈ V |
| 52 | 51 | mptex 7243 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺)))
∈ V |
| 53 | 52 | rnex 7932 |
. . . . . . . 8
⊢ ran
(𝑧 ∈ 𝑎 ↦
(((0g‘𝐺)
+ 𝑧) −
(0g‘𝐺)))
∈ V |
| 54 | 50, 37, 53 | ovmpoa 7588 |
. . . . . . 7
⊢
(((0g‘𝐺) ∈ 𝑋 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → ((0g‘𝐺) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺)))) |
| 55 | 43, 44, 54 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → ((0g‘𝐺) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺)))) |
| 56 | 1 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧 ∈ 𝑎) → 𝐺 ∈ Grp) |
| 57 | | slwsubg 19628 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (𝑃 pSyl 𝐺) → 𝑎 ∈ (SubGrp‘𝐺)) |
| 58 | 57 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → 𝑎 ∈ (SubGrp‘𝐺)) |
| 59 | 6 | subgss 19145 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ (SubGrp‘𝐺) → 𝑎 ⊆ 𝑋) |
| 60 | 58, 59 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → 𝑎 ⊆ 𝑋) |
| 61 | 60 | sselda 3983 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧 ∈ 𝑎) → 𝑧 ∈ 𝑋) |
| 62 | 6, 13, 41 | grplid 18985 |
. . . . . . . . . . . . 13
⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺) + 𝑧) = 𝑧) |
| 63 | 56, 61, 62 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧 ∈ 𝑎) → ((0g‘𝐺) + 𝑧) = 𝑧) |
| 64 | 63 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧 ∈ 𝑎) → (((0g‘𝐺) + 𝑧) −
(0g‘𝐺)) =
(𝑧 −
(0g‘𝐺))) |
| 65 | 6, 41, 14 | grpsubid1 19043 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (𝑧 −
(0g‘𝐺)) =
𝑧) |
| 66 | 56, 61, 65 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧 ∈ 𝑎) → (𝑧 −
(0g‘𝐺)) =
𝑧) |
| 67 | 64, 66 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑧 ∈ 𝑎) → (((0g‘𝐺) + 𝑧) −
(0g‘𝐺)) =
𝑧) |
| 68 | 67 | mpteq2dva 5242 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺))) =
(𝑧 ∈ 𝑎 ↦ 𝑧)) |
| 69 | | mptresid 6069 |
. . . . . . . . 9
⊢ ( I
↾ 𝑎) = (𝑧 ∈ 𝑎 ↦ 𝑧) |
| 70 | 68, 69 | eqtr4di 2795 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺))) =
( I ↾ 𝑎)) |
| 71 | 70 | rneqd 5949 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → ran (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺))) =
ran ( I ↾ 𝑎)) |
| 72 | | rnresi 6093 |
. . . . . . 7
⊢ ran ( I
↾ 𝑎) = 𝑎 |
| 73 | 71, 72 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → ran (𝑧 ∈ 𝑎 ↦ (((0g‘𝐺) + 𝑧) −
(0g‘𝐺))) =
𝑎) |
| 74 | 55, 73 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → ((0g‘𝐺) ⊕ 𝑎) = 𝑎) |
| 75 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((𝑐 + 𝑧) − 𝑐) ∈ V |
| 76 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑤 = ((𝑐 + 𝑧) − 𝑐) → (𝑏 + 𝑤) = (𝑏 + ((𝑐 + 𝑧) − 𝑐))) |
| 77 | 76 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (𝑤 = ((𝑐 + 𝑧) − 𝑐) → ((𝑏 + 𝑤) − 𝑏) = ((𝑏 + ((𝑐 + 𝑧) − 𝑐)) − 𝑏)) |
| 78 | 75, 77 | abrexco 7264 |
. . . . . . . . 9
⊢ {𝑢 ∣ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = ((𝑐 + 𝑧) − 𝑐)}𝑢 = ((𝑏 + 𝑤) − 𝑏)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + ((𝑐 + 𝑧) − 𝑐)) − 𝑏)} |
| 79 | | simprr 773 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑐 ∈ 𝑋) |
| 80 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑎 ∈ (𝑃 pSyl 𝐺)) |
| 81 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎) |
| 82 | | simpl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → 𝑥 = 𝑐) |
| 83 | 82 | oveq1d 7446 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = (𝑐 + 𝑧)) |
| 84 | 83, 82 | oveq12d 7449 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → ((𝑥 + 𝑧) − 𝑥) = ((𝑐 + 𝑧) − 𝑐)) |
| 85 | 81, 84 | mpteq12dv 5233 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐))) |
| 86 | 85 | rneqd 5949 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = 𝑐 ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐))) |
| 87 | 51 | mptex 7243 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐)) ∈ V |
| 88 | 87 | rnex 7932 |
. . . . . . . . . . . . . 14
⊢ ran
(𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐)) ∈ V |
| 89 | 86, 37, 88 | ovmpoa 7588 |
. . . . . . . . . . . . 13
⊢ ((𝑐 ∈ 𝑋 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → (𝑐 ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐))) |
| 90 | 79, 80, 89 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐))) |
| 91 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐)) = (𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐)) |
| 92 | 91 | rnmpt 5968 |
. . . . . . . . . . . 12
⊢ ran
(𝑧 ∈ 𝑎 ↦ ((𝑐 + 𝑧) − 𝑐)) = {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = ((𝑐 + 𝑧) − 𝑐)} |
| 93 | 90, 92 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) = {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = ((𝑐 + 𝑧) − 𝑐)}) |
| 94 | 93 | rexeqdv 3327 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = ((𝑏 + 𝑤) − 𝑏) ↔ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = ((𝑐 + 𝑧) − 𝑐)}𝑢 = ((𝑏 + 𝑤) − 𝑏))) |
| 95 | 94 | abbidv 2808 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = ((𝑏 + 𝑤) − 𝑏)} = {𝑢 ∣ ∃𝑤 ∈ {𝑣 ∣ ∃𝑧 ∈ 𝑎 𝑣 = ((𝑐 + 𝑧) − 𝑐)}𝑢 = ((𝑏 + 𝑤) − 𝑏)}) |
| 96 | 40 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝐺 ∈ Grp) |
| 97 | 96 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝐺 ∈ Grp) |
| 98 | | simprl 771 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → 𝑏 ∈ 𝑋) |
| 99 | 6, 13 | grpcl 18959 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋) → (𝑏 + 𝑐) ∈ 𝑋) |
| 100 | 96, 98, 79, 99 | syl3anc 1373 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑏 + 𝑐) ∈ 𝑋) |
| 101 | 100 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (𝑏 + 𝑐) ∈ 𝑋) |
| 102 | 61 | adantlr 715 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑧 ∈ 𝑋) |
| 103 | 6, 13 | grpcl 18959 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺 ∈ Grp ∧ (𝑏 + 𝑐) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑏 + 𝑐) + 𝑧) ∈ 𝑋) |
| 104 | 97, 101, 102, 103 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → ((𝑏 + 𝑐) + 𝑧) ∈ 𝑋) |
| 105 | 79 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑐 ∈ 𝑋) |
| 106 | 98 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → 𝑏 ∈ 𝑋) |
| 107 | 6, 13, 14 | grpsubsub4 19051 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Grp ∧ (((𝑏 + 𝑐) + 𝑧) ∈ 𝑋 ∧ 𝑐 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((((𝑏 + 𝑐) + 𝑧) − 𝑐) − 𝑏) = (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))) |
| 108 | 97, 104, 105, 106, 107 | syl13anc 1374 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → ((((𝑏 + 𝑐) + 𝑧) − 𝑐) − 𝑏) = (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))) |
| 109 | 6, 13 | grpass 18960 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺 ∈ Grp ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧))) |
| 110 | 97, 106, 105, 102, 109 | syl13anc 1374 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → ((𝑏 + 𝑐) + 𝑧) = (𝑏 + (𝑐 + 𝑧))) |
| 111 | 110 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (((𝑏 + 𝑐) + 𝑧) − 𝑐) = ((𝑏 + (𝑐 + 𝑧)) − 𝑐)) |
| 112 | 6, 13 | grpcl 18959 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺 ∈ Grp ∧ 𝑐 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑐 + 𝑧) ∈ 𝑋) |
| 113 | 97, 105, 102, 112 | syl3anc 1373 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (𝑐 + 𝑧) ∈ 𝑋) |
| 114 | 6, 13, 14 | grpaddsubass 19048 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺 ∈ Grp ∧ (𝑏 ∈ 𝑋 ∧ (𝑐 + 𝑧) ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + (𝑐 + 𝑧)) − 𝑐) = (𝑏 + ((𝑐 + 𝑧) − 𝑐))) |
| 115 | 97, 106, 113, 105, 114 | syl13anc 1374 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → ((𝑏 + (𝑐 + 𝑧)) − 𝑐) = (𝑏 + ((𝑐 + 𝑧) − 𝑐))) |
| 116 | 111, 115 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (((𝑏 + 𝑐) + 𝑧) − 𝑐) = (𝑏 + ((𝑐 + 𝑧) − 𝑐))) |
| 117 | 116 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → ((((𝑏 + 𝑐) + 𝑧) − 𝑐) − 𝑏) = ((𝑏 + ((𝑐 + 𝑧) − 𝑐)) − 𝑏)) |
| 118 | 108, 117 | eqtr3d 2779 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐)) = ((𝑏 + ((𝑐 + 𝑧) − 𝑐)) − 𝑏)) |
| 119 | 118 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑎) → (𝑢 = (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐)) ↔ 𝑢 = ((𝑏 + ((𝑐 + 𝑧) − 𝑐)) − 𝑏))) |
| 120 | 119 | rexbidva 3177 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (∃𝑧 ∈ 𝑎 𝑢 = (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐)) ↔ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + ((𝑐 + 𝑧) − 𝑐)) − 𝑏))) |
| 121 | 120 | abbidv 2808 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = ((𝑏 + ((𝑐 + 𝑧) − 𝑐)) − 𝑏)}) |
| 122 | 78, 95, 121 | 3eqtr4a 2803 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = ((𝑏 + 𝑤) − 𝑏)} = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))}) |
| 123 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏)) |
| 124 | 123 | rnmpt 5968 |
. . . . . . . 8
⊢ ran
(𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏)) = {𝑢 ∣ ∃𝑤 ∈ (𝑐 ⊕ 𝑎)𝑢 = ((𝑏 + 𝑤) − 𝑏)} |
| 125 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))) = (𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))) |
| 126 | 125 | rnmpt 5968 |
. . . . . . . 8
⊢ ran
(𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))) = {𝑢 ∣ ∃𝑧 ∈ 𝑎 𝑢 = (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))} |
| 127 | 122, 124,
126 | 3eqtr4g 2802 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏)) = ran (𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐)))) |
| 128 | 39 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ⊕ :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺)) |
| 129 | 128, 79, 80 | fovcdmd 7605 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑐 ⊕ 𝑎) ∈ (𝑃 pSyl 𝐺)) |
| 130 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → 𝑦 = (𝑐 ⊕ 𝑎)) |
| 131 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → 𝑥 = 𝑏) |
| 132 | 131 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑥 + 𝑧) = (𝑏 + 𝑧)) |
| 133 | 132, 131 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → ((𝑥 + 𝑧) − 𝑥) = ((𝑏 + 𝑧) − 𝑏)) |
| 134 | 130, 133 | mpteq12dv 5233 |
. . . . . . . . . . 11
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑧) − 𝑏))) |
| 135 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑤 → (𝑏 + 𝑧) = (𝑏 + 𝑤)) |
| 136 | 135 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑤 → ((𝑏 + 𝑧) − 𝑏) = ((𝑏 + 𝑤) − 𝑏)) |
| 137 | 136 | cbvmptv 5255 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑧) − 𝑏)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏)) |
| 138 | 134, 137 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏))) |
| 139 | 138 | rneqd 5949 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑏 ∧ 𝑦 = (𝑐 ⊕ 𝑎)) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏))) |
| 140 | | ovex 7464 |
. . . . . . . . . . 11
⊢ (𝑐 ⊕ 𝑎) ∈ V |
| 141 | 140 | mptex 7243 |
. . . . . . . . . 10
⊢ (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏)) ∈ V |
| 142 | 141 | rnex 7932 |
. . . . . . . . 9
⊢ ran
(𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏)) ∈ V |
| 143 | 139, 37, 142 | ovmpoa 7588 |
. . . . . . . 8
⊢ ((𝑏 ∈ 𝑋 ∧ (𝑐 ⊕ 𝑎) ∈ (𝑃 pSyl 𝐺)) → (𝑏 ⊕ (𝑐 ⊕ 𝑎)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏))) |
| 144 | 98, 129, 143 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → (𝑏 ⊕ (𝑐 ⊕ 𝑎)) = ran (𝑤 ∈ (𝑐 ⊕ 𝑎) ↦ ((𝑏 + 𝑤) − 𝑏))) |
| 145 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑦 = 𝑎) |
| 146 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → 𝑥 = (𝑏 + 𝑐)) |
| 147 | 146 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑥 + 𝑧) = ((𝑏 + 𝑐) + 𝑧)) |
| 148 | 147, 146 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → ((𝑥 + 𝑧) − 𝑥) = (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))) |
| 149 | 145, 148 | mpteq12dv 5233 |
. . . . . . . . . 10
⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐)))) |
| 150 | 149 | rneqd 5949 |
. . . . . . . . 9
⊢ ((𝑥 = (𝑏 + 𝑐) ∧ 𝑦 = 𝑎) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐)))) |
| 151 | 51 | mptex 7243 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))) ∈ V |
| 152 | 151 | rnex 7932 |
. . . . . . . . 9
⊢ ran
(𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐))) ∈ V |
| 153 | 150, 37, 152 | ovmpoa 7588 |
. . . . . . . 8
⊢ (((𝑏 + 𝑐) ∈ 𝑋 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → ((𝑏 + 𝑐) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐)))) |
| 154 | 100, 80, 153 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + 𝑐) ⊕ 𝑎) = ran (𝑧 ∈ 𝑎 ↦ (((𝑏 + 𝑐) + 𝑧) − (𝑏 + 𝑐)))) |
| 155 | 127, 144,
154 | 3eqtr4rd 2788 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) ∧ (𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋)) → ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))) |
| 156 | 155 | ralrimivva 3202 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))) |
| 157 | 74, 156 | jca 511 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑃 pSyl 𝐺)) → (((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))) |
| 158 | 157 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑎 ∈ (𝑃 pSyl 𝐺)(((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))) |
| 159 | 39, 158 | jca 511 |
. 2
⊢ (𝜑 → ( ⊕ :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺) ∧ ∀𝑎 ∈ (𝑃 pSyl 𝐺)(((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎))))) |
| 160 | 6, 13, 41 | isga 19309 |
. 2
⊢ ( ⊕ ∈
(𝐺 GrpAct (𝑃 pSyl 𝐺)) ↔ ((𝐺 ∈ Grp ∧ (𝑃 pSyl 𝐺) ∈ V) ∧ ( ⊕ :(𝑋 × (𝑃 pSyl 𝐺))⟶(𝑃 pSyl 𝐺) ∧ ∀𝑎 ∈ (𝑃 pSyl 𝐺)(((0g‘𝐺) ⊕ 𝑎) = 𝑎 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ((𝑏 + 𝑐) ⊕ 𝑎) = (𝑏 ⊕ (𝑐 ⊕ 𝑎)))))) |
| 161 | 3, 159, 160 | sylanbrc 583 |
1
⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺))) |