| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 2 | | nsgqusf1o.s |
. . . . . . . . . 10
⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} |
| 3 | 2 | reqabi 3460 |
. . . . . . . . 9
⊢ (ℎ ∈ 𝑆 ↔ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) |
| 4 | | nsgqusf1o.b |
. . . . . . . . . . . 12
⊢ 𝐵 = (Base‘𝐺) |
| 5 | | nsgqusf1o.t |
. . . . . . . . . . . 12
⊢ 𝑇 = (SubGrp‘𝑄) |
| 6 | | nsgqusf1o.1 |
. . . . . . . . . . . 12
⊢ ≤ =
(le‘(toInc‘𝑆)) |
| 7 | | nsgqusf1o.2 |
. . . . . . . . . . . 12
⊢ ≲ =
(le‘(toInc‘𝑇)) |
| 8 | | nsgqusf1o.q |
. . . . . . . . . . . 12
⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) |
| 9 | | nsgqusf1o.p |
. . . . . . . . . . . 12
⊢ ⊕ =
(LSSum‘𝐺) |
| 10 | | nsgqusf1o.e |
. . . . . . . . . . . 12
⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 11 | | nsgqusf1o.f |
. . . . . . . . . . . 12
⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) |
| 12 | | nsgqusf1o.n |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) |
| 13 | 4, 2, 5, 6, 7, 8, 9, 10, 11, 12 | nsgqusf1olem1 33441 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇) |
| 14 | 13 | anasss 466 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇) |
| 15 | 14, 5 | eleqtrdi 2851 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄)) |
| 16 | 3, 15 | sylan2b 594 |
. . . . . . . 8
⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄)) |
| 17 | 16 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄)) |
| 18 | 1, 17 | eqeltrd 2841 |
. . . . . 6
⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄)) |
| 19 | 18 | r19.29an 3158 |
. . . . 5
⊢ ((𝜑 ∧ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄)) |
| 20 | | sseq2 4010 |
. . . . . . . 8
⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑁 ⊆ ℎ ↔ 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})) |
| 21 | 12 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺)) |
| 22 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄)) |
| 23 | 4, 8, 9, 21, 22 | nsgmgclem 33439 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺)) |
| 24 | 5 | eleq2i 2833 |
. . . . . . . . 9
⊢ (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrp‘𝑄)) |
| 25 | | nsgsubg 19176 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) |
| 26 | 12, 25 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺)) |
| 27 | 4 | subgss 19145 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵) |
| 28 | 26, 27 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ⊆ 𝐵) |
| 29 | 28 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ 𝐵) |
| 30 | 26 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺)) |
| 31 | 9 | grplsmid 33432 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁) |
| 32 | 30, 31 | sylancom 588 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁) |
| 33 | 24 | biimpi 216 |
. . . . . . . . . . . . 13
⊢ (𝑓 ∈ 𝑇 → 𝑓 ∈ (SubGrp‘𝑄)) |
| 34 | 8 | nsgqus0 33438 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓) |
| 35 | 12, 33, 34 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ 𝑓) |
| 36 | 35 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓) |
| 37 | 32, 36 | eqeltrd 2841 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓) |
| 38 | 29, 37 | ssrabdv 4074 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) |
| 39 | 24, 38 | sylan2br 595 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) |
| 40 | 20, 23, 39 | elrabd 3694 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}) |
| 41 | 40, 2 | eleqtrrdi 2852 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ 𝑆) |
| 42 | | mpteq1 5235 |
. . . . . . . . 9
⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))) |
| 43 | 42 | rneqd 5949 |
. . . . . . . 8
⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))) |
| 44 | 43 | eqeq2d 2748 |
. . . . . . 7
⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))) |
| 45 | 44 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))) |
| 46 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(Base‘𝑄) =
(Base‘𝑄) |
| 47 | 46 | subgss 19145 |
. . . . . . . . . . . . . 14
⊢ (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄)) |
| 48 | 47 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄)) |
| 49 | 48 | sselda 3983 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑖 ∈ (Base‘𝑄)) |
| 50 | 8 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))) |
| 51 | 4 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝐵 = (Base‘𝐺)) |
| 52 | | ovexd 7466 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐺 ~QG 𝑁) ∈ V) |
| 53 | | subgrcl 19149 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
| 54 | 26, 53 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 55 | 54 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝐺 ∈ Grp) |
| 56 | 50, 51, 52, 55 | qusbas 17590 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄)) |
| 57 | 49, 56 | eleqtrrd 2844 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁))) |
| 58 | | elqsi 8810 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)) → ∃𝑥 ∈ 𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁)) |
| 59 | 57, 58 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → ∃𝑥 ∈ 𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁)) |
| 60 | | sneq 4636 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑥 → {𝑎} = {𝑥}) |
| 61 | 60 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑥 → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) |
| 62 | 61 | eleq1d 2826 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑥 → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) |
| 63 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ 𝐵) |
| 64 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁)) |
| 65 | 26 | ad4antr 732 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺)) |
| 66 | 4, 9, 65, 63 | quslsm 33433 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁)) |
| 67 | 64, 66 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} ⊕ 𝑁)) |
| 68 | | simpllr 776 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 ∈ 𝑓) |
| 69 | 67, 68 | eqeltrrd 2842 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} ⊕ 𝑁) ∈ 𝑓) |
| 70 | 62, 63, 69 | elrabd 3694 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) |
| 71 | 70, 67 | jca 511 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} ⊕ 𝑁))) |
| 72 | 71 | expl 457 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → ((𝑥 ∈ 𝐵 ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} ⊕ 𝑁)))) |
| 73 | 72 | reximdv2 3164 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (∃𝑥 ∈ 𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁) → ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁))) |
| 74 | 59, 73 | mpd 15 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)) |
| 75 | | simplr 769 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) |
| 76 | 62 | elrab 3692 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↔ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) |
| 77 | 75, 76 | sylib 218 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) |
| 78 | | simpllr 776 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} ⊕ 𝑁)) |
| 79 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → ({𝑥} ⊕ 𝑁) ∈ 𝑓) |
| 80 | 78, 79 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 ∈ 𝑓) |
| 81 | 80 | anasss 466 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) → 𝑖 ∈ 𝑓) |
| 82 | 81 | adantllr 719 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) → 𝑖 ∈ 𝑓) |
| 83 | 77, 82 | mpdan 687 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓) |
| 84 | 83 | r19.29an 3158 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓) |
| 85 | 74, 84 | impbida 801 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁))) |
| 86 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) |
| 87 | 86 | elrnmpt 5969 |
. . . . . . . . 9
⊢ (𝑖 ∈ V → (𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁))) |
| 88 | 87 | elv 3485 |
. . . . . . . 8
⊢ (𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)) |
| 89 | 85, 88 | bitr4di 289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ 𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))) |
| 90 | 89 | eqrdv 2735 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))) |
| 91 | 41, 45, 90 | rspcedvd 3624 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
| 92 | 19, 91 | impbida 801 |
. . . 4
⊢ (𝜑 → (∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄))) |
| 93 | 92 | abbidv 2808 |
. . 3
⊢ (𝜑 → {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))} = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)}) |
| 94 | 10 | rnmpt 5968 |
. . 3
⊢ ran 𝐸 = {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))} |
| 95 | | abid1 2878 |
. . 3
⊢
(SubGrp‘𝑄) =
{𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)} |
| 96 | 93, 94, 95 | 3eqtr4g 2802 |
. 2
⊢ (𝜑 → ran 𝐸 = (SubGrp‘𝑄)) |
| 97 | 96, 5 | eqtr4di 2795 |
1
⊢ (𝜑 → ran 𝐸 = 𝑇) |