Proof of Theorem lshpkrlem4
| Step | Hyp | Ref
| Expression |
| 1 | | simp3l 1202 |
. . . 4
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → 𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍))) |
| 2 | 1 | oveq2d 7447 |
. . 3
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → (𝑙 · 𝑢) = (𝑙 · (𝑟 + ((𝐺‘𝑢) · 𝑍)))) |
| 3 | | simp3r 1203 |
. . 3
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍))) |
| 4 | 2, 3 | oveq12d 7449 |
. 2
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) = ((𝑙 · (𝑟 + ((𝐺‘𝑢) · 𝑍))) + (𝑠 + ((𝐺‘𝑣) · 𝑍)))) |
| 5 | | simpl1 1192 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → 𝜑) |
| 6 | | lshpkrlem.w |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ LVec) |
| 7 | | lveclmod 21105 |
. . . . . . . 8
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) |
| 8 | 5, 6, 7 | 3syl 18 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → 𝑊 ∈ LMod) |
| 9 | | simpl2 1193 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → 𝑙 ∈ 𝐾) |
| 10 | | simpr2 1196 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → 𝑟 ∈ 𝑉) |
| 11 | | simpl3 1194 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → 𝑢 ∈ 𝑉) |
| 12 | | lshpkrlem.v |
. . . . . . . . . 10
⊢ 𝑉 = (Base‘𝑊) |
| 13 | | lshpkrlem.a |
. . . . . . . . . 10
⊢ + =
(+g‘𝑊) |
| 14 | | lshpkrlem.n |
. . . . . . . . . 10
⊢ 𝑁 = (LSpan‘𝑊) |
| 15 | | lshpkrlem.p |
. . . . . . . . . 10
⊢ ⊕ =
(LSSum‘𝑊) |
| 16 | | lshpkrlem.h |
. . . . . . . . . 10
⊢ 𝐻 = (LSHyp‘𝑊) |
| 17 | 6 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → 𝑊 ∈ LVec) |
| 18 | | lshpkrlem.u |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈 ∈ 𝐻) |
| 19 | 18 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → 𝑈 ∈ 𝐻) |
| 20 | | lshpkrlem.z |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑍 ∈ 𝑉) |
| 21 | 20 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → 𝑍 ∈ 𝑉) |
| 22 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → 𝑢 ∈ 𝑉) |
| 23 | | lshpkrlem.e |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) |
| 24 | 23 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) |
| 25 | | lshpkrlem.d |
. . . . . . . . . 10
⊢ 𝐷 = (Scalar‘𝑊) |
| 26 | | lshpkrlem.k |
. . . . . . . . . 10
⊢ 𝐾 = (Base‘𝐷) |
| 27 | | lshpkrlem.t |
. . . . . . . . . 10
⊢ · = (
·𝑠 ‘𝑊) |
| 28 | | lshpkrlem.o |
. . . . . . . . . 10
⊢ 0 =
(0g‘𝐷) |
| 29 | | lshpkrlem.g |
. . . . . . . . . 10
⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) |
| 30 | 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 27, 28, 29 | lshpkrlem2 39112 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (𝐺‘𝑢) ∈ 𝐾) |
| 31 | 5, 11, 30 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (𝐺‘𝑢) ∈ 𝐾) |
| 32 | 5, 20 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → 𝑍 ∈ 𝑉) |
| 33 | 12, 25, 27, 26 | lmodvscl 20876 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ (𝐺‘𝑢) ∈ 𝐾 ∧ 𝑍 ∈ 𝑉) → ((𝐺‘𝑢) · 𝑍) ∈ 𝑉) |
| 34 | 8, 31, 32, 33 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → ((𝐺‘𝑢) · 𝑍) ∈ 𝑉) |
| 35 | 12, 13, 25, 27, 26 | lmodvsdi 20883 |
. . . . . . 7
⊢ ((𝑊 ∈ LMod ∧ (𝑙 ∈ 𝐾 ∧ 𝑟 ∈ 𝑉 ∧ ((𝐺‘𝑢) · 𝑍) ∈ 𝑉)) → (𝑙 · (𝑟 + ((𝐺‘𝑢) · 𝑍))) = ((𝑙 · 𝑟) + (𝑙 · ((𝐺‘𝑢) · 𝑍)))) |
| 36 | 8, 9, 10, 34, 35 | syl13anc 1374 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (𝑙 · (𝑟 + ((𝐺‘𝑢) · 𝑍))) = ((𝑙 · 𝑟) + (𝑙 · ((𝐺‘𝑢) · 𝑍)))) |
| 37 | | eqid 2737 |
. . . . . . . . 9
⊢
(.r‘𝐷) = (.r‘𝐷) |
| 38 | 12, 25, 27, 26, 37 | lmodvsass 20885 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ (𝑙 ∈ 𝐾 ∧ (𝐺‘𝑢) ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) = (𝑙 · ((𝐺‘𝑢) · 𝑍))) |
| 39 | 8, 9, 31, 32, 38 | syl13anc 1374 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) = (𝑙 · ((𝐺‘𝑢) · 𝑍))) |
| 40 | 39 | oveq2d 7447 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → ((𝑙 · 𝑟) + ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍)) = ((𝑙 · 𝑟) + (𝑙 · ((𝐺‘𝑢) · 𝑍)))) |
| 41 | 36, 40 | eqtr4d 2780 |
. . . . 5
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (𝑙 · (𝑟 + ((𝐺‘𝑢) · 𝑍))) = ((𝑙 · 𝑟) + ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍))) |
| 42 | 41 | oveq1d 7446 |
. . . 4
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → ((𝑙 · (𝑟 + ((𝐺‘𝑢) · 𝑍))) + (𝑠 + ((𝐺‘𝑣) · 𝑍))) = (((𝑙 · 𝑟) + ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍)) + (𝑠 + ((𝐺‘𝑣) · 𝑍)))) |
| 43 | 12, 25, 27, 26 | lmodvscl 20876 |
. . . . . . 7
⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ 𝐾 ∧ 𝑟 ∈ 𝑉) → (𝑙 · 𝑟) ∈ 𝑉) |
| 44 | 8, 9, 10, 43 | syl3anc 1373 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (𝑙 · 𝑟) ∈ 𝑉) |
| 45 | 25, 26, 37 | lmodmcl 20871 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ 𝐾 ∧ (𝐺‘𝑢) ∈ 𝐾) → (𝑙(.r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾) |
| 46 | 8, 9, 31, 45 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (𝑙(.r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾) |
| 47 | 12, 25, 27, 26 | lmodvscl 20876 |
. . . . . . 7
⊢ ((𝑊 ∈ LMod ∧ (𝑙(.r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾 ∧ 𝑍 ∈ 𝑉) → ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) ∈ 𝑉) |
| 48 | 8, 46, 32, 47 | syl3anc 1373 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) ∈ 𝑉) |
| 49 | | simpr3 1197 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → 𝑠 ∈ 𝑉) |
| 50 | | simpr1 1195 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → 𝑣 ∈ 𝑉) |
| 51 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) |
| 52 | 18 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑈 ∈ 𝐻) |
| 53 | 20 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑍 ∈ 𝑉) |
| 54 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) |
| 55 | 23 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) |
| 56 | 12, 13, 14, 15, 16, 51, 52, 53, 54, 55, 25, 26, 27, 28, 29 | lshpkrlem2 39112 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
| 57 | 5, 50, 56 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (𝐺‘𝑣) ∈ 𝐾) |
| 58 | 12, 25, 27, 26 | lmodvscl 20876 |
. . . . . . 7
⊢ ((𝑊 ∈ LMod ∧ (𝐺‘𝑣) ∈ 𝐾 ∧ 𝑍 ∈ 𝑉) → ((𝐺‘𝑣) · 𝑍) ∈ 𝑉) |
| 59 | 8, 57, 32, 58 | syl3anc 1373 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → ((𝐺‘𝑣) · 𝑍) ∈ 𝑉) |
| 60 | 12, 13 | lmod4 20910 |
. . . . . 6
⊢ ((𝑊 ∈ LMod ∧ ((𝑙 · 𝑟) ∈ 𝑉 ∧ ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) ∈ 𝑉) ∧ (𝑠 ∈ 𝑉 ∧ ((𝐺‘𝑣) · 𝑍) ∈ 𝑉)) → (((𝑙 · 𝑟) + ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍)) + (𝑠 + ((𝐺‘𝑣) · 𝑍))) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) + ((𝐺‘𝑣) · 𝑍)))) |
| 61 | 8, 44, 48, 49, 59, 60 | syl122anc 1381 |
. . . . 5
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (((𝑙 · 𝑟) + ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍)) + (𝑠 + ((𝐺‘𝑣) · 𝑍))) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) + ((𝐺‘𝑣) · 𝑍)))) |
| 62 | | eqid 2737 |
. . . . . . . 8
⊢
(+g‘𝐷) = (+g‘𝐷) |
| 63 | 12, 13, 25, 27, 26, 62 | lmodvsdir 20884 |
. . . . . . 7
⊢ ((𝑊 ∈ LMod ∧ ((𝑙(.r‘𝐷)(𝐺‘𝑢)) ∈ 𝐾 ∧ (𝐺‘𝑣) ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍) = (((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) + ((𝐺‘𝑣) · 𝑍))) |
| 64 | 8, 46, 57, 32, 63 | syl13anc 1374 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍) = (((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) + ((𝐺‘𝑣) · 𝑍))) |
| 65 | 64 | oveq2d 7447 |
. . . . 5
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍)) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍) + ((𝐺‘𝑣) · 𝑍)))) |
| 66 | 61, 65 | eqtr4d 2780 |
. . . 4
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → (((𝑙 · 𝑟) + ((𝑙(.r‘𝐷)(𝐺‘𝑢)) · 𝑍)) + (𝑠 + ((𝐺‘𝑣) · 𝑍))) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍))) |
| 67 | 42, 66 | eqtrd 2777 |
. . 3
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉)) → ((𝑙 · (𝑟 + ((𝐺‘𝑢) · 𝑍))) + (𝑠 + ((𝐺‘𝑣) · 𝑍))) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍))) |
| 68 | 67 | 3adant3 1133 |
. 2
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → ((𝑙 · (𝑟 + ((𝐺‘𝑢) · 𝑍))) + (𝑠 + ((𝐺‘𝑣) · 𝑍))) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍))) |
| 69 | 4, 68 | eqtrd 2777 |
1
⊢ (((𝜑 ∧ 𝑙 ∈ 𝐾 ∧ 𝑢 ∈ 𝑉) ∧ (𝑣 ∈ 𝑉 ∧ 𝑟 ∈ 𝑉 ∧ 𝑠 ∈ 𝑉) ∧ (𝑢 = (𝑟 + ((𝐺‘𝑢) · 𝑍)) ∧ 𝑣 = (𝑠 + ((𝐺‘𝑣) · 𝑍)))) → ((𝑙 · 𝑢) + 𝑣) = (((𝑙 · 𝑟) + 𝑠) + (((𝑙(.r‘𝐷)(𝐺‘𝑢))(+g‘𝐷)(𝐺‘𝑣)) · 𝑍))) |