| Step | Hyp | Ref
| Expression |
| 1 | | nsgqusf1o.n |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) |
| 2 | | nsgqusf1o.q |
. . . . . 6
⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) |
| 3 | 2 | qusgrp 19204 |
. . . . 5
⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp) |
| 4 | 1, 3 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑄 ∈ Grp) |
| 5 | 4 | ad2antrr 726 |
. . 3
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → 𝑄 ∈ Grp) |
| 6 | | nsgqusf1o.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝐺) |
| 7 | 6 | subgss 19145 |
. . . . . . . . 9
⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ⊆ 𝐵) |
| 8 | 7 | ad2antlr 727 |
. . . . . . . 8
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ ⊆ 𝐵) |
| 9 | 8 | sselda 3983 |
. . . . . . 7
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → 𝑥 ∈ 𝐵) |
| 10 | | ovex 7464 |
. . . . . . . 8
⊢ (𝐺 ~QG 𝑁) ∈ V |
| 11 | 10 | ecelqsi 8813 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐵 → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁))) |
| 12 | 9, 11 | syl 17 |
. . . . . 6
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁))) |
| 13 | | nsgqusf1o.p |
. . . . . . 7
⊢ ⊕ =
(LSSum‘𝐺) |
| 14 | | nsgsubg 19176 |
. . . . . . . . 9
⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) |
| 15 | 1, 14 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺)) |
| 16 | 15 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → 𝑁 ∈ (SubGrp‘𝐺)) |
| 17 | 6, 13, 16, 9 | quslsm 33433 |
. . . . . 6
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁)) |
| 18 | 2 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))) |
| 19 | 6 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
| 20 | | ovexd 7466 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ~QG 𝑁) ∈ V) |
| 21 | | subgrcl 19149 |
. . . . . . . . 9
⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
| 22 | 15, 21 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 23 | 18, 19, 20, 22 | qusbas 17590 |
. . . . . . 7
⊢ (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄)) |
| 24 | 23 | ad3antrrr 730 |
. . . . . 6
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄)) |
| 25 | 12, 17, 24 | 3eltr3d 2855 |
. . . . 5
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄)) |
| 26 | 25 | ralrimiva 3146 |
. . . 4
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ∀𝑥 ∈ ℎ ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄)) |
| 27 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) |
| 28 | 27 | rnmptss 7143 |
. . . 4
⊢
(∀𝑥 ∈
ℎ ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄)) |
| 29 | 26, 28 | syl 17 |
. . 3
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄)) |
| 30 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑥((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) |
| 31 | | ovexd 7466 |
. . . 4
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ({𝑥} ⊕ 𝑁) ∈ V) |
| 32 | | eqid 2737 |
. . . . . . 7
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 33 | 32 | subg0cl 19152 |
. . . . . 6
⊢ (ℎ ∈ (SubGrp‘𝐺) →
(0g‘𝐺)
∈ ℎ) |
| 34 | 33 | ne0d 4342 |
. . . . 5
⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ≠ ∅) |
| 35 | 34 | ad2antlr 727 |
. . . 4
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ ≠ ∅) |
| 36 | 30, 31, 27, 35 | rnmptn0 6264 |
. . 3
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ≠ ∅) |
| 37 | | nfmpt1 5250 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) |
| 38 | 37 | nfrn 5963 |
. . . . . . 7
⊢
Ⅎ𝑥ran
(𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) |
| 39 | 38 | nfel2 2924 |
. . . . . 6
⊢
Ⅎ𝑥 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) |
| 40 | 30, 39 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑥(((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 41 | 38 | nfel2 2924 |
. . . . . . 7
⊢
Ⅎ𝑥(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) |
| 42 | 38, 41 | nfralw 3311 |
. . . . . 6
⊢
Ⅎ𝑥∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) |
| 43 | 38 | nfel2 2924 |
. . . . . 6
⊢
Ⅎ𝑥((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) |
| 44 | 42, 43 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑥(∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 45 | | sneq 4636 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧}) |
| 46 | 45 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑧 → ({𝑥} ⊕ 𝑁) = ({𝑧} ⊕ 𝑁)) |
| 47 | 46 | cbvmptv 5255 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑧 ∈ ℎ ↦ ({𝑧} ⊕ 𝑁)) |
| 48 | | simp-4r 784 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺)) |
| 49 | 48 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺)) |
| 50 | | simp-4r 784 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑥 ∈ ℎ) |
| 51 | | simplr 769 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑦 ∈ ℎ) |
| 52 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 53 | 52 | subgcl 19154 |
. . . . . . . . . . . . 13
⊢ ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ ℎ ∧ 𝑦 ∈ ℎ) → (𝑥(+g‘𝐺)𝑦) ∈ ℎ) |
| 54 | 49, 50, 51, 53 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑥(+g‘𝐺)𝑦) ∈ ℎ) |
| 55 | | sneq 4636 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → {𝑧} = {(𝑥(+g‘𝐺)𝑦)}) |
| 56 | 55 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → ({𝑧} ⊕ 𝑁) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁)) |
| 57 | 56 | eqeq2d 2748 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → ((𝑖(+g‘𝑄)𝑗) = ({𝑧} ⊕ 𝑁) ↔ (𝑖(+g‘𝑄)𝑗) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁))) |
| 58 | 57 | adantl 481 |
. . . . . . . . . . . 12
⊢
((((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) ∧ 𝑧 = (𝑥(+g‘𝐺)𝑦)) → ((𝑖(+g‘𝑄)𝑗) = ({𝑧} ⊕ 𝑁) ↔ (𝑖(+g‘𝑄)𝑗) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁))) |
| 59 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 = ({𝑥} ⊕ 𝑁)) |
| 60 | 17 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁)) |
| 61 | 59, 60 | eqtr4d 2780 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁)) |
| 62 | 61 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁)) |
| 63 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑗 = ({𝑦} ⊕ 𝑁)) |
| 64 | 1 | ad4antr 732 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺)) |
| 65 | 64 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺)) |
| 66 | 65, 14 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺)) |
| 67 | 49, 7 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ℎ ⊆ 𝐵) |
| 68 | 67, 51 | sseldd 3984 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑦 ∈ 𝐵) |
| 69 | 6, 13, 66, 68 | quslsm 33433 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → [𝑦](𝐺 ~QG 𝑁) = ({𝑦} ⊕ 𝑁)) |
| 70 | 63, 69 | eqtr4d 2780 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑗 = [𝑦](𝐺 ~QG 𝑁)) |
| 71 | 62, 70 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) = ([𝑥](𝐺 ~QG 𝑁)(+g‘𝑄)[𝑦](𝐺 ~QG 𝑁))) |
| 72 | 9 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ 𝐵) |
| 73 | 72 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑥 ∈ 𝐵) |
| 74 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(+g‘𝑄) = (+g‘𝑄) |
| 75 | 2, 6, 52, 74 | qusadd 19206 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ([𝑥](𝐺 ~QG 𝑁)(+g‘𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(+g‘𝐺)𝑦)](𝐺 ~QG 𝑁)) |
| 76 | 65, 73, 68, 75 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ([𝑥](𝐺 ~QG 𝑁)(+g‘𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(+g‘𝐺)𝑦)](𝐺 ~QG 𝑁)) |
| 77 | 67, 54 | sseldd 3984 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
| 78 | 6, 13, 66, 77 | quslsm 33433 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → [(𝑥(+g‘𝐺)𝑦)](𝐺 ~QG 𝑁) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁)) |
| 79 | 71, 76, 78 | 3eqtrd 2781 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁)) |
| 80 | 54, 58, 79 | rspcedvd 3624 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ∃𝑧 ∈ ℎ (𝑖(+g‘𝑄)𝑗) = ({𝑧} ⊕ 𝑁)) |
| 81 | | ovexd 7466 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) ∈ V) |
| 82 | 47, 80, 81 | elrnmptd 5974 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 83 | 82 | adantllr 719 |
. . . . . . . . 9
⊢
((((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 84 | | sneq 4636 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) |
| 85 | 84 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → ({𝑥} ⊕ 𝑁) = ({𝑦} ⊕ 𝑁)) |
| 86 | 85 | cbvmptv 5255 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑦 ∈ ℎ ↦ ({𝑦} ⊕ 𝑁)) |
| 87 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ ({𝑦} ⊕ 𝑁) ∈ V |
| 88 | 86, 87 | elrnmpti 5973 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑦 ∈ ℎ 𝑗 = ({𝑦} ⊕ 𝑁)) |
| 89 | 88 | biimpi 216 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → ∃𝑦 ∈ ℎ 𝑗 = ({𝑦} ⊕ 𝑁)) |
| 90 | 89 | adantl 481 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ∃𝑦 ∈ ℎ 𝑗 = ({𝑦} ⊕ 𝑁)) |
| 91 | 83, 90 | r19.29a 3162 |
. . . . . . . 8
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → (𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 92 | 91 | ralrimiva 3146 |
. . . . . . 7
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 93 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(invg‘𝐺) = (invg‘𝐺) |
| 94 | 93 | subginvcl 19153 |
. . . . . . . . . 10
⊢ ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ ℎ) |
| 95 | 94 | ad5ant24 761 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝐺)‘𝑥) ∈ ℎ) |
| 96 | | simpr 484 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → 𝑦 = ((invg‘𝐺)‘𝑥)) |
| 97 | 96 | sneqd 4638 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → {𝑦} = {((invg‘𝐺)‘𝑥)}) |
| 98 | 97 | oveq1d 7446 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → ({𝑦} ⊕ 𝑁) = ({((invg‘𝐺)‘𝑥)} ⊕ 𝑁)) |
| 99 | 8 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ℎ ⊆ 𝐵) |
| 100 | 94 | ad4ant24 754 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ ℎ) |
| 101 | 99, 100 | sseldd 3984 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ 𝐵) |
| 102 | 6, 13, 16, 101 | quslsm 33433 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg‘𝐺)‘𝑥)} ⊕ 𝑁)) |
| 103 | 102 | ad2antrr 726 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg‘𝐺)‘𝑥)} ⊕ 𝑁)) |
| 104 | 98, 103 | eqtr4d 2780 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → ({𝑦} ⊕ 𝑁) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁)) |
| 105 | 104 | eqeq2d 2748 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → (((invg‘𝑄)‘𝑖) = ({𝑦} ⊕ 𝑁) ↔ ((invg‘𝑄)‘𝑖) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))) |
| 106 | 61 | fveq2d 6910 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) = ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁))) |
| 107 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(invg‘𝑄) = (invg‘𝑄) |
| 108 | 2, 6, 93, 107 | qusinv 19208 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝐵) → ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁)) |
| 109 | 64, 72, 108 | syl2anc 584 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁)) |
| 110 | 106, 109 | eqtrd 2777 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁)) |
| 111 | 95, 105, 110 | rspcedvd 3624 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ∃𝑦 ∈ ℎ ((invg‘𝑄)‘𝑖) = ({𝑦} ⊕ 𝑁)) |
| 112 | | fvexd 6921 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) ∈ V) |
| 113 | 86, 111, 112 | elrnmptd 5974 |
. . . . . . 7
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 114 | 92, 113 | jca 511 |
. . . . . 6
⊢
(((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → (∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))) |
| 115 | 114 | adantllr 719 |
. . . . 5
⊢
((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → (∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))) |
| 116 | | ovex 7464 |
. . . . . . . 8
⊢ ({𝑥} ⊕ 𝑁) ∈ V |
| 117 | 27, 116 | elrnmpti 5973 |
. . . . . . 7
⊢ (𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ ℎ 𝑖 = ({𝑥} ⊕ 𝑁)) |
| 118 | 117 | biimpi 216 |
. . . . . 6
⊢ (𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → ∃𝑥 ∈ ℎ 𝑖 = ({𝑥} ⊕ 𝑁)) |
| 119 | 118 | adantl 481 |
. . . . 5
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ∃𝑥 ∈ ℎ 𝑖 = ({𝑥} ⊕ 𝑁)) |
| 120 | 40, 44, 115, 119 | r19.29af2 3267 |
. . . 4
⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → (∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))) |
| 121 | 120 | ralrimiva 3146 |
. . 3
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ∀𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))) |
| 122 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝑄) =
(Base‘𝑄) |
| 123 | 122, 74, 107 | issubg2 19159 |
. . . 4
⊢ (𝑄 ∈ Grp → (ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄) ↔ (ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄) ∧ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ≠ ∅ ∧ ∀𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))))) |
| 124 | 123 | biimpar 477 |
. . 3
⊢ ((𝑄 ∈ Grp ∧ (ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄) ∧ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ≠ ∅ ∧ ∀𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄)) |
| 125 | 5, 29, 36, 121, 124 | syl13anc 1374 |
. 2
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄)) |
| 126 | | nsgqusf1o.t |
. 2
⊢ 𝑇 = (SubGrp‘𝑄) |
| 127 | 125, 126 | eleqtrrdi 2852 |
1
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇) |