| Step | Hyp | Ref
| Expression |
| 1 | | elrgspn.r |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ Ring) |
| 2 | | elrgspn.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
| 3 | 2 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
| 4 | | elrgspn.a |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| 5 | | elrgspn.n |
. . . . . 6
⊢ 𝑁 = (RingSpan‘𝑅) |
| 6 | 5 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) |
| 7 | | eqidd 2738 |
. . . . 5
⊢ (𝜑 → (𝑁‘𝐴) = (𝑁‘𝐴)) |
| 8 | 1, 3, 4, 6, 7 | rgspncl 20613 |
. . . 4
⊢ (𝜑 → (𝑁‘𝐴) ∈ (SubRing‘𝑅)) |
| 9 | 2 | subrgss 20572 |
. . . 4
⊢ ((𝑁‘𝐴) ∈ (SubRing‘𝑅) → (𝑁‘𝐴) ⊆ 𝐵) |
| 10 | 8, 9 | syl 17 |
. . 3
⊢ (𝜑 → (𝑁‘𝐴) ⊆ 𝐵) |
| 11 | 10 | sselda 3983 |
. 2
⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘𝐴)) → 𝑋 ∈ 𝐵) |
| 12 | | simpr 484 |
. . . 4
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) → 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) |
| 13 | | eqid 2737 |
. . . . . 6
⊢
(0g‘𝑅) = (0g‘𝑅) |
| 14 | 1 | ringcmnd 20281 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ CMnd) |
| 15 | 14 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝑅 ∈ CMnd) |
| 16 | 2 | fvexi 6920 |
. . . . . . . . . 10
⊢ 𝐵 ∈ V |
| 17 | 16 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ V) |
| 18 | 17, 4 | ssexd 5324 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ V) |
| 19 | 18 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝐴 ∈ V) |
| 20 | | wrdexg 14562 |
. . . . . . 7
⊢ (𝐴 ∈ V → Word 𝐴 ∈ V) |
| 21 | 19, 20 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → Word 𝐴 ∈ V) |
| 22 | | elrgspn.x |
. . . . . . . 8
⊢ · =
(.g‘𝑅) |
| 23 | 1 | ringgrpd 20239 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ Grp) |
| 24 | 23 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑤 ∈ Word 𝐴) → 𝑅 ∈ Grp) |
| 25 | | zex 12622 |
. . . . . . . . . . 11
⊢ ℤ
∈ V |
| 26 | 25 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → ℤ ∈ V) |
| 27 | | breq1 5146 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑔 → (𝑓 finSupp 0 ↔ 𝑔 finSupp 0)) |
| 28 | | elrgspn.f |
. . . . . . . . . . . . . 14
⊢ 𝐹 = {𝑓 ∈ (ℤ ↑m Word
𝐴) ∣ 𝑓 finSupp 0} |
| 29 | 27, 28 | elrab2 3695 |
. . . . . . . . . . . . 13
⊢ (𝑔 ∈ 𝐹 ↔ (𝑔 ∈ (ℤ ↑m Word
𝐴) ∧ 𝑔 finSupp 0)) |
| 30 | 29 | biimpi 216 |
. . . . . . . . . . . 12
⊢ (𝑔 ∈ 𝐹 → (𝑔 ∈ (ℤ ↑m Word
𝐴) ∧ 𝑔 finSupp 0)) |
| 31 | 30 | simpld 494 |
. . . . . . . . . . 11
⊢ (𝑔 ∈ 𝐹 → 𝑔 ∈ (ℤ ↑m Word
𝐴)) |
| 32 | 31 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝑔 ∈ (ℤ ↑m Word
𝐴)) |
| 33 | 21, 26, 32 | elmaprd 32689 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝑔:Word 𝐴⟶ℤ) |
| 34 | 33 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑤 ∈ Word 𝐴) → (𝑔‘𝑤) ∈ ℤ) |
| 35 | | elrgspn.m |
. . . . . . . . . . . 12
⊢ 𝑀 = (mulGrp‘𝑅) |
| 36 | 35 | ringmgp 20236 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
| 37 | 1, 36 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ Mnd) |
| 38 | 37 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑤 ∈ Word 𝐴) → 𝑀 ∈ Mnd) |
| 39 | | sswrd 14560 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ 𝐵 → Word 𝐴 ⊆ Word 𝐵) |
| 40 | 4, 39 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → Word 𝐴 ⊆ Word 𝐵) |
| 41 | 40 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → Word 𝐴 ⊆ Word 𝐵) |
| 42 | 41 | sselda 3983 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑤 ∈ Word 𝐴) → 𝑤 ∈ Word 𝐵) |
| 43 | 35, 2 | mgpbas 20142 |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝑀) |
| 44 | 43 | gsumwcl 18852 |
. . . . . . . . 9
⊢ ((𝑀 ∈ Mnd ∧ 𝑤 ∈ Word 𝐵) → (𝑀 Σg 𝑤) ∈ 𝐵) |
| 45 | 38, 42, 44 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑤 ∈ Word 𝐴) → (𝑀 Σg 𝑤) ∈ 𝐵) |
| 46 | 2, 22, 24, 34, 45 | mulgcld 19114 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑤 ∈ Word 𝐴) → ((𝑔‘𝑤) · (𝑀 Σg 𝑤)) ∈ 𝐵) |
| 47 | 46 | fmpttd 7135 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))):Word 𝐴⟶𝐵) |
| 48 | 33 | feqmptd 6977 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝑔 = (𝑤 ∈ Word 𝐴 ↦ (𝑔‘𝑤))) |
| 49 | 30 | simprd 495 |
. . . . . . . . 9
⊢ (𝑔 ∈ 𝐹 → 𝑔 finSupp 0) |
| 50 | 49 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → 𝑔 finSupp 0) |
| 51 | 48, 50 | eqbrtrrd 5167 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (𝑤 ∈ Word 𝐴 ↦ (𝑔‘𝑤)) finSupp 0) |
| 52 | 2, 13, 22 | mulg0 19092 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐵 → (0 · 𝑦) = (0g‘𝑅)) |
| 53 | 52 | adantl 481 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑦 ∈ 𝐵) → (0 · 𝑦) = (0g‘𝑅)) |
| 54 | | fvexd 6921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (0g‘𝑅) ∈ V) |
| 55 | 51, 53, 34, 45, 54 | fsuppssov1 9424 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))) finSupp
(0g‘𝑅)) |
| 56 | 2, 13, 15, 21, 47, 55 | gsumcl 19933 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ 𝐹) → (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤)))) ∈ 𝐵) |
| 57 | 56 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) → (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤)))) ∈ 𝐵) |
| 58 | 12, 57 | eqeltrd 2841 |
. . 3
⊢ (((𝜑 ∧ 𝑔 ∈ 𝐹) ∧ 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) → 𝑋 ∈ 𝐵) |
| 59 | 58 | r19.29an 3158 |
. 2
⊢ ((𝜑 ∧ ∃𝑔 ∈ 𝐹 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) → 𝑋 ∈ 𝐵) |
| 60 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Ring) |
| 61 | 4 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐴 ⊆ 𝐵) |
| 62 | | fveq1 6905 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑖 → (ℎ‘𝑤) = (𝑖‘𝑤)) |
| 63 | 62 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (ℎ = 𝑖 → ((ℎ‘𝑤) · (𝑀 Σg 𝑤)) = ((𝑖‘𝑤) · (𝑀 Σg 𝑤))) |
| 64 | 63 | mpteq2dv 5244 |
. . . . . . . . 9
⊢ (ℎ = 𝑖 → (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))) = (𝑤 ∈ Word 𝐴 ↦ ((𝑖‘𝑤) · (𝑀 Σg 𝑤)))) |
| 65 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑣 → (𝑖‘𝑤) = (𝑖‘𝑣)) |
| 66 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑣 → (𝑀 Σg 𝑤) = (𝑀 Σg 𝑣)) |
| 67 | 65, 66 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑣 → ((𝑖‘𝑤) · (𝑀 Σg 𝑤)) = ((𝑖‘𝑣) · (𝑀 Σg 𝑣))) |
| 68 | 67 | cbvmptv 5255 |
. . . . . . . . 9
⊢ (𝑤 ∈ Word 𝐴 ↦ ((𝑖‘𝑤) · (𝑀 Σg 𝑤))) = (𝑣 ∈ Word 𝐴 ↦ ((𝑖‘𝑣) · (𝑀 Σg 𝑣))) |
| 69 | 64, 68 | eqtrdi 2793 |
. . . . . . . 8
⊢ (ℎ = 𝑖 → (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))) = (𝑣 ∈ Word 𝐴 ↦ ((𝑖‘𝑣) · (𝑀 Σg 𝑣)))) |
| 70 | 69 | oveq2d 7447 |
. . . . . . 7
⊢ (ℎ = 𝑖 → (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤)))) = (𝑅 Σg (𝑣 ∈ Word 𝐴 ↦ ((𝑖‘𝑣) · (𝑀 Σg 𝑣))))) |
| 71 | 70 | cbvmptv 5255 |
. . . . . 6
⊢ (ℎ ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))))) = (𝑖 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ Word 𝐴 ↦ ((𝑖‘𝑣) · (𝑀 Σg 𝑣))))) |
| 72 | 71 | rneqi 5948 |
. . . . 5
⊢ ran
(ℎ ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))))) = ran (𝑖 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ Word 𝐴 ↦ ((𝑖‘𝑣) · (𝑀 Σg 𝑣))))) |
| 73 | 2, 35, 22, 5, 28, 60, 61, 72 | elrgspnlem4 33249 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝐴) = ran (ℎ ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤)))))) |
| 74 | 73 | eleq2d 2827 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (𝑁‘𝐴) ↔ 𝑋 ∈ ran (ℎ ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))))))) |
| 75 | | fveq1 6905 |
. . . . . . . . 9
⊢ (ℎ = 𝑔 → (ℎ‘𝑤) = (𝑔‘𝑤)) |
| 76 | 75 | oveq1d 7446 |
. . . . . . . 8
⊢ (ℎ = 𝑔 → ((ℎ‘𝑤) · (𝑀 Σg 𝑤)) = ((𝑔‘𝑤) · (𝑀 Σg 𝑤))) |
| 77 | 76 | mpteq2dv 5244 |
. . . . . . 7
⊢ (ℎ = 𝑔 → (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))) = (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤)))) |
| 78 | 77 | oveq2d 7447 |
. . . . . 6
⊢ (ℎ = 𝑔 → (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤)))) = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) |
| 79 | 78 | cbvmptv 5255 |
. . . . 5
⊢ (ℎ ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))))) = (𝑔 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) |
| 80 | 79 | elrnmpt 5969 |
. . . 4
⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ ran (ℎ ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))))) ↔ ∃𝑔 ∈ 𝐹 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤)))))) |
| 81 | 80 | adantl 481 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ ran (ℎ ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((ℎ‘𝑤) · (𝑀 Σg 𝑤))))) ↔ ∃𝑔 ∈ 𝐹 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤)))))) |
| 82 | 74, 81 | bitrd 279 |
. 2
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (𝑁‘𝐴) ↔ ∃𝑔 ∈ 𝐹 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤)))))) |
| 83 | 11, 59, 82 | bibiad 840 |
1
⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝐴) ↔ ∃𝑔 ∈ 𝐹 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤)))))) |