Proof of Theorem lmodfopne
| Step | Hyp | Ref
 | Expression | 
| 1 |   | lmodfopne.t | 
. . . . . 6
⊢  · = (
·sf ‘𝑊) | 
| 2 |   | lmodfopne.a | 
. . . . . 6
⊢  + =
(+𝑓‘𝑊) | 
| 3 |   | lmodfopne.v | 
. . . . . 6
⊢ 𝑉 = (Base‘𝑊) | 
| 4 |   | lmodfopne.s | 
. . . . . 6
⊢ 𝑆 = (Scalar‘𝑊) | 
| 5 |   | lmodfopne.k | 
. . . . . 6
⊢ 𝐾 = (Base‘𝑆) | 
| 6 |   | lmodfopne.0 | 
. . . . . 6
⊢  0 =
(0g‘𝑆) | 
| 7 |   | lmodfopne.1 | 
. . . . . 6
⊢  1 =
(1r‘𝑆) | 
| 8 | 1, 2, 3, 4, 5, 6, 7 | lmodfopnelem2 13881 | 
. . . . 5
⊢ ((𝑊 ∈ LMod ∧ + = · )
→ ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) | 
| 9 |   | simpll 527 | 
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → 𝑊 ∈ LMod) | 
| 10 |   | simpl 109 | 
. . . . . . . . 9
⊢ (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → 0 ∈ 𝑉) | 
| 11 | 10 | adantl 277 | 
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → 0 ∈ 𝑉) | 
| 12 |   | eqid 2196 | 
. . . . . . . . . 10
⊢
(0g‘𝑊) = (0g‘𝑊) | 
| 13 | 3, 12 | lmod0vcl 13873 | 
. . . . . . . . 9
⊢ (𝑊 ∈ LMod →
(0g‘𝑊)
∈ 𝑉) | 
| 14 | 13 | ad2antrr 488 | 
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) →
(0g‘𝑊)
∈ 𝑉) | 
| 15 |   | eqid 2196 | 
. . . . . . . . . 10
⊢
(+g‘𝑊) = (+g‘𝑊) | 
| 16 | 3, 15, 2 | plusfvalg 13006 | 
. . . . . . . . 9
⊢ ((𝑊 ∈ LMod ∧ 0 ∈ 𝑉 ∧
(0g‘𝑊)
∈ 𝑉) → ( 0 +
(0g‘𝑊)) =
( 0
(+g‘𝑊)(0g‘𝑊))) | 
| 17 | 16 | eqcomd 2202 | 
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 0 ∈ 𝑉 ∧
(0g‘𝑊)
∈ 𝑉) → ( 0
(+g‘𝑊)(0g‘𝑊)) = ( 0 + (0g‘𝑊))) | 
| 18 | 9, 11, 14, 17 | syl3anc 1249 | 
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 0 (+g‘𝑊)(0g‘𝑊)) = ( 0 + (0g‘𝑊))) | 
| 19 |   | oveq 5928 | 
. . . . . . . 8
⊢ ( + = · →
( 0 +
(0g‘𝑊)) =
( 0 ·
(0g‘𝑊))) | 
| 20 | 19 | ad2antlr 489 | 
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 0 + (0g‘𝑊)) = ( 0 ·
(0g‘𝑊))) | 
| 21 | 18, 20 | eqtrd 2229 | 
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 0 (+g‘𝑊)(0g‘𝑊)) = ( 0 ·
(0g‘𝑊))) | 
| 22 |   | lmodgrp 13850 | 
. . . . . . . 8
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | 
| 23 | 22 | adantr 276 | 
. . . . . . 7
⊢ ((𝑊 ∈ LMod ∧ + = · )
→ 𝑊 ∈
Grp) | 
| 24 | 3, 15, 12 | grprid 13164 | 
. . . . . . 7
⊢ ((𝑊 ∈ Grp ∧ 0 ∈ 𝑉) → ( 0 (+g‘𝑊)(0g‘𝑊)) = 0 ) | 
| 25 | 23, 10, 24 | syl2an 289 | 
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 0 (+g‘𝑊)(0g‘𝑊)) = 0 ) | 
| 26 | 4, 5, 6 | lmod0cl 13870 | 
. . . . . . . . 9
⊢ (𝑊 ∈ LMod → 0 ∈ 𝐾) | 
| 27 | 26 | ad2antrr 488 | 
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → 0 ∈ 𝐾) | 
| 28 |   | eqid 2196 | 
. . . . . . . . 9
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) | 
| 29 | 3, 4, 5, 1, 28 | scafvalg 13863 | 
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 0 ∈ 𝐾 ∧
(0g‘𝑊)
∈ 𝑉) → ( 0 ·
(0g‘𝑊)) =
( 0 (
·𝑠 ‘𝑊)(0g‘𝑊))) | 
| 30 | 9, 27, 14, 29 | syl3anc 1249 | 
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 0 ·
(0g‘𝑊)) =
( 0 (
·𝑠 ‘𝑊)(0g‘𝑊))) | 
| 31 | 26 | ancli 323 | 
. . . . . . . . 9
⊢ (𝑊 ∈ LMod → (𝑊 ∈ LMod ∧ 0 ∈ 𝐾)) | 
| 32 | 31 | ad2antrr 488 | 
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → (𝑊 ∈ LMod ∧ 0 ∈ 𝐾)) | 
| 33 | 4, 28, 5, 12 | lmodvs0 13878 | 
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 0 ∈ 𝐾) → ( 0 (
·𝑠 ‘𝑊)(0g‘𝑊)) = (0g‘𝑊)) | 
| 34 | 32, 33 | syl 14 | 
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 0 (
·𝑠 ‘𝑊)(0g‘𝑊)) = (0g‘𝑊)) | 
| 35 |   | simpr 110 | 
. . . . . . . . . 10
⊢ (( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → 1 ∈ 𝑉) | 
| 36 | 3, 15, 12 | grprid 13164 | 
. . . . . . . . . 10
⊢ ((𝑊 ∈ Grp ∧ 1 ∈ 𝑉) → ( 1 (+g‘𝑊)(0g‘𝑊)) = 1 ) | 
| 37 | 23, 35, 36 | syl2an 289 | 
. . . . . . . . 9
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 1 (+g‘𝑊)(0g‘𝑊)) = 1 ) | 
| 38 | 4, 5, 7 | lmod1cl 13871 | 
. . . . . . . . . . . 12
⊢ (𝑊 ∈ LMod → 1 ∈ 𝐾) | 
| 39 | 38 | ad2antrr 488 | 
. . . . . . . . . . 11
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → 1 ∈ 𝐾) | 
| 40 | 35 | adantl 277 | 
. . . . . . . . . . 11
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → 1 ∈ 𝑉) | 
| 41 | 3, 4, 5, 1, 28 | scafvalg 13863 | 
. . . . . . . . . . 11
⊢ ((𝑊 ∈ LMod ∧ 1 ∈ 𝐾 ∧ 1 ∈ 𝑉) → ( 1 · 1 ) = ( 1 (
·𝑠 ‘𝑊) 1 )) | 
| 42 | 9, 39, 40, 41 | syl3anc 1249 | 
. . . . . . . . . 10
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 1 · 1 ) = ( 1 (
·𝑠 ‘𝑊) 1 )) | 
| 43 | 3, 4, 28, 7 | lmodvs1 13872 | 
. . . . . . . . . . 11
⊢ ((𝑊 ∈ LMod ∧ 1 ∈ 𝑉) → ( 1 (
·𝑠 ‘𝑊) 1 ) = 1 ) | 
| 44 | 43 | ad2ant2rl 511 | 
. . . . . . . . . 10
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 1 (
·𝑠 ‘𝑊) 1 ) = 1 ) | 
| 45 | 42, 44 | eqtrd 2229 | 
. . . . . . . . 9
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 1 · 1 ) = 1 ) | 
| 46 |   | oveq 5928 | 
. . . . . . . . . . . 12
⊢ ( + = · →
( 1 + 1 ) = ( 1 · 1
)) | 
| 47 | 46 | eqcomd 2202 | 
. . . . . . . . . . 11
⊢ ( + = · →
( 1 · 1 ) = ( 1 + 1
)) | 
| 48 | 47 | ad2antlr 489 | 
. . . . . . . . . 10
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 1 · 1 ) = ( 1 + 1 )) | 
| 49 | 3, 15, 2 | plusfvalg 13006 | 
. . . . . . . . . . 11
⊢ ((𝑊 ∈ LMod ∧ 1 ∈ 𝑉 ∧ 1 ∈ 𝑉) → ( 1 + 1 ) = ( 1 (+g‘𝑊) 1 )) | 
| 50 | 9, 40, 40, 49 | syl3anc 1249 | 
. . . . . . . . . 10
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 1 + 1 ) = ( 1 (+g‘𝑊) 1 )) | 
| 51 | 48, 50 | eqtrd 2229 | 
. . . . . . . . 9
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 1 · 1 ) = ( 1 (+g‘𝑊) 1 )) | 
| 52 | 37, 45, 51 | 3eqtr2d 2235 | 
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 1 (+g‘𝑊)(0g‘𝑊)) = ( 1 (+g‘𝑊) 1 )) | 
| 53 | 22 | ad2antrr 488 | 
. . . . . . . . 9
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → 𝑊 ∈ Grp) | 
| 54 | 3, 15 | grplcan 13194 | 
. . . . . . . . 9
⊢ ((𝑊 ∈ Grp ∧
((0g‘𝑊)
∈ 𝑉 ∧ 1 ∈ 𝑉 ∧ 1 ∈ 𝑉)) → (( 1 (+g‘𝑊)(0g‘𝑊)) = ( 1 (+g‘𝑊) 1 ) ↔
(0g‘𝑊) =
1
)) | 
| 55 | 53, 14, 40, 40, 54 | syl13anc 1251 | 
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → (( 1 (+g‘𝑊)(0g‘𝑊)) = ( 1 (+g‘𝑊) 1 ) ↔
(0g‘𝑊) =
1
)) | 
| 56 | 52, 55 | mpbid 147 | 
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) →
(0g‘𝑊) =
1
) | 
| 57 | 30, 34, 56 | 3eqtrd 2233 | 
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → ( 0 ·
(0g‘𝑊)) =
1
) | 
| 58 | 21, 25, 57 | 3eqtr3rd 2238 | 
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ + = · )
∧ ( 0
∈ 𝑉 ∧ 1 ∈ 𝑉)) → 1 = 0 ) | 
| 59 | 8, 58 | mpdan 421 | 
. . . 4
⊢ ((𝑊 ∈ LMod ∧ + = · )
→ 1
= 0
) | 
| 60 | 59 | ex 115 | 
. . 3
⊢ (𝑊 ∈ LMod → ( + = · →
1 = 0
)) | 
| 61 | 60 | necon3d 2411 | 
. 2
⊢ (𝑊 ∈ LMod → ( 1 ≠ 0 → + ≠ ·
)) | 
| 62 | 61 | imp 124 | 
1
⊢ ((𝑊 ∈ LMod ∧ 1 ≠ 0 ) →
+ ≠
·
) |