| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gsumwun.p |
⊢ + = ( +g ‘ 𝑀 ) |
| 2 |
|
gsumwun.m |
⊢ ( 𝜑 → 𝑀 ∈ CMnd ) |
| 3 |
|
gsumwun.e |
⊢ ( 𝜑 → 𝐸 ∈ ( SubMnd ‘ 𝑀 ) ) |
| 4 |
|
gsumwun.f |
⊢ ( 𝜑 → 𝐹 ∈ ( SubMnd ‘ 𝑀 ) ) |
| 5 |
|
gsumwun.w |
⊢ ( 𝜑 → 𝑊 ∈ Word ( 𝐸 ∪ 𝐹 ) ) |
| 6 |
|
oveq2 |
⊢ ( 𝑣 = ∅ → ( 𝑀 Σg 𝑣 ) = ( 𝑀 Σg ∅ ) ) |
| 7 |
6
|
eqeq1d |
⊢ ( 𝑣 = ∅ → ( ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ( 𝑀 Σg ∅ ) = ( 𝑒 + 𝑓 ) ) ) |
| 8 |
7
|
2rexbidv |
⊢ ( 𝑣 = ∅ → ( ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg ∅ ) = ( 𝑒 + 𝑓 ) ) ) |
| 9 |
8
|
imbi2d |
⊢ ( 𝑣 = ∅ → ( ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ) ↔ ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg ∅ ) = ( 𝑒 + 𝑓 ) ) ) ) |
| 10 |
|
oveq2 |
⊢ ( 𝑣 = 𝑤 → ( 𝑀 Σg 𝑣 ) = ( 𝑀 Σg 𝑤 ) ) |
| 11 |
10
|
eqeq1d |
⊢ ( 𝑣 = 𝑤 → ( ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ) |
| 12 |
11
|
2rexbidv |
⊢ ( 𝑣 = 𝑤 → ( ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ) |
| 13 |
12
|
imbi2d |
⊢ ( 𝑣 = 𝑤 → ( ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ) ↔ ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ) ) |
| 14 |
|
oveq1 |
⊢ ( 𝑒 = 𝑖 → ( 𝑒 + 𝑓 ) = ( 𝑖 + 𝑓 ) ) |
| 15 |
14
|
eqeq2d |
⊢ ( 𝑒 = 𝑖 → ( ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ( 𝑀 Σg 𝑣 ) = ( 𝑖 + 𝑓 ) ) ) |
| 16 |
|
oveq2 |
⊢ ( 𝑓 = 𝑗 → ( 𝑖 + 𝑓 ) = ( 𝑖 + 𝑗 ) ) |
| 17 |
16
|
eqeq2d |
⊢ ( 𝑓 = 𝑗 → ( ( 𝑀 Σg 𝑣 ) = ( 𝑖 + 𝑓 ) ↔ ( 𝑀 Σg 𝑣 ) = ( 𝑖 + 𝑗 ) ) ) |
| 18 |
15 17
|
cbvrex2vw |
⊢ ( ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑖 + 𝑗 ) ) |
| 19 |
|
oveq2 |
⊢ ( 𝑣 = ( 𝑤 ++ 〈“ 𝑥 ”〉 ) → ( 𝑀 Σg 𝑣 ) = ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) ) |
| 20 |
19
|
eqeq1d |
⊢ ( 𝑣 = ( 𝑤 ++ 〈“ 𝑥 ”〉 ) → ( ( 𝑀 Σg 𝑣 ) = ( 𝑖 + 𝑗 ) ↔ ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) ) |
| 21 |
20
|
2rexbidv |
⊢ ( 𝑣 = ( 𝑤 ++ 〈“ 𝑥 ”〉 ) → ( ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑖 + 𝑗 ) ↔ ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) ) |
| 22 |
18 21
|
bitrid |
⊢ ( 𝑣 = ( 𝑤 ++ 〈“ 𝑥 ”〉 ) → ( ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) ) |
| 23 |
22
|
imbi2d |
⊢ ( 𝑣 = ( 𝑤 ++ 〈“ 𝑥 ”〉 ) → ( ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ) ↔ ( 𝜑 → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) ) ) |
| 24 |
|
oveq2 |
⊢ ( 𝑣 = 𝑊 → ( 𝑀 Σg 𝑣 ) = ( 𝑀 Σg 𝑊 ) ) |
| 25 |
24
|
eqeq1d |
⊢ ( 𝑣 = 𝑊 → ( ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ( 𝑀 Σg 𝑊 ) = ( 𝑒 + 𝑓 ) ) ) |
| 26 |
25
|
2rexbidv |
⊢ ( 𝑣 = 𝑊 → ( ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ↔ ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑊 ) = ( 𝑒 + 𝑓 ) ) ) |
| 27 |
26
|
imbi2d |
⊢ ( 𝑣 = 𝑊 → ( ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑣 ) = ( 𝑒 + 𝑓 ) ) ↔ ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑊 ) = ( 𝑒 + 𝑓 ) ) ) ) |
| 28 |
|
oveq1 |
⊢ ( 𝑒 = ( 0g ‘ 𝑀 ) → ( 𝑒 + 𝑓 ) = ( ( 0g ‘ 𝑀 ) + 𝑓 ) ) |
| 29 |
28
|
eqeq2d |
⊢ ( 𝑒 = ( 0g ‘ 𝑀 ) → ( ( 𝑀 Σg ∅ ) = ( 𝑒 + 𝑓 ) ↔ ( 𝑀 Σg ∅ ) = ( ( 0g ‘ 𝑀 ) + 𝑓 ) ) ) |
| 30 |
|
oveq2 |
⊢ ( 𝑓 = ( 0g ‘ 𝑀 ) → ( ( 0g ‘ 𝑀 ) + 𝑓 ) = ( ( 0g ‘ 𝑀 ) + ( 0g ‘ 𝑀 ) ) ) |
| 31 |
30
|
eqeq2d |
⊢ ( 𝑓 = ( 0g ‘ 𝑀 ) → ( ( 𝑀 Σg ∅ ) = ( ( 0g ‘ 𝑀 ) + 𝑓 ) ↔ ( 𝑀 Σg ∅ ) = ( ( 0g ‘ 𝑀 ) + ( 0g ‘ 𝑀 ) ) ) ) |
| 32 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
| 33 |
32
|
subm0cl |
⊢ ( 𝐸 ∈ ( SubMnd ‘ 𝑀 ) → ( 0g ‘ 𝑀 ) ∈ 𝐸 ) |
| 34 |
3 33
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑀 ) ∈ 𝐸 ) |
| 35 |
32
|
subm0cl |
⊢ ( 𝐹 ∈ ( SubMnd ‘ 𝑀 ) → ( 0g ‘ 𝑀 ) ∈ 𝐹 ) |
| 36 |
4 35
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑀 ) ∈ 𝐹 ) |
| 37 |
32
|
gsum0 |
⊢ ( 𝑀 Σg ∅ ) = ( 0g ‘ 𝑀 ) |
| 38 |
2
|
cmnmndd |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
| 39 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
| 40 |
39 32
|
mndidcl |
⊢ ( 𝑀 ∈ Mnd → ( 0g ‘ 𝑀 ) ∈ ( Base ‘ 𝑀 ) ) |
| 41 |
39 1 32
|
mndlid |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( 0g ‘ 𝑀 ) ∈ ( Base ‘ 𝑀 ) ) → ( ( 0g ‘ 𝑀 ) + ( 0g ‘ 𝑀 ) ) = ( 0g ‘ 𝑀 ) ) |
| 42 |
38 40 41
|
syl2anc2 |
⊢ ( 𝜑 → ( ( 0g ‘ 𝑀 ) + ( 0g ‘ 𝑀 ) ) = ( 0g ‘ 𝑀 ) ) |
| 43 |
37 42
|
eqtr4id |
⊢ ( 𝜑 → ( 𝑀 Σg ∅ ) = ( ( 0g ‘ 𝑀 ) + ( 0g ‘ 𝑀 ) ) ) |
| 44 |
29 31 34 36 43
|
2rspcedvdw |
⊢ ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg ∅ ) = ( 𝑒 + 𝑓 ) ) |
| 45 |
|
oveq1 |
⊢ ( 𝑖 = ( 𝑒 + 𝑥 ) → ( 𝑖 + 𝑗 ) = ( ( 𝑒 + 𝑥 ) + 𝑗 ) ) |
| 46 |
45
|
eqeq2d |
⊢ ( 𝑖 = ( 𝑒 + 𝑥 ) → ( ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ↔ ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( ( 𝑒 + 𝑥 ) + 𝑗 ) ) ) |
| 47 |
|
oveq2 |
⊢ ( 𝑗 = 𝑓 → ( ( 𝑒 + 𝑥 ) + 𝑗 ) = ( ( 𝑒 + 𝑥 ) + 𝑓 ) ) |
| 48 |
47
|
eqeq2d |
⊢ ( 𝑗 = 𝑓 → ( ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( ( 𝑒 + 𝑥 ) + 𝑗 ) ↔ ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( ( 𝑒 + 𝑥 ) + 𝑓 ) ) ) |
| 49 |
3
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐸 ) → 𝐸 ∈ ( SubMnd ‘ 𝑀 ) ) |
| 50 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐸 ) → 𝑒 ∈ 𝐸 ) |
| 51 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐸 ) → 𝑥 ∈ 𝐸 ) |
| 52 |
1 49 50 51
|
submcld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐸 ) → ( 𝑒 + 𝑥 ) ∈ 𝐸 ) |
| 53 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐸 ) → 𝑓 ∈ 𝐹 ) |
| 54 |
38
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → 𝑀 ∈ Mnd ) |
| 55 |
39
|
submss |
⊢ ( 𝐸 ∈ ( SubMnd ‘ 𝑀 ) → 𝐸 ⊆ ( Base ‘ 𝑀 ) ) |
| 56 |
3 55
|
syl |
⊢ ( 𝜑 → 𝐸 ⊆ ( Base ‘ 𝑀 ) ) |
| 57 |
39
|
submss |
⊢ ( 𝐹 ∈ ( SubMnd ‘ 𝑀 ) → 𝐹 ⊆ ( Base ‘ 𝑀 ) ) |
| 58 |
4 57
|
syl |
⊢ ( 𝜑 → 𝐹 ⊆ ( Base ‘ 𝑀 ) ) |
| 59 |
56 58
|
unssd |
⊢ ( 𝜑 → ( 𝐸 ∪ 𝐹 ) ⊆ ( Base ‘ 𝑀 ) ) |
| 60 |
|
sswrd |
⊢ ( ( 𝐸 ∪ 𝐹 ) ⊆ ( Base ‘ 𝑀 ) → Word ( 𝐸 ∪ 𝐹 ) ⊆ Word ( Base ‘ 𝑀 ) ) |
| 61 |
59 60
|
syl |
⊢ ( 𝜑 → Word ( 𝐸 ∪ 𝐹 ) ⊆ Word ( Base ‘ 𝑀 ) ) |
| 62 |
61
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) → 𝑤 ∈ Word ( Base ‘ 𝑀 ) ) |
| 63 |
62
|
ad4antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → 𝑤 ∈ Word ( Base ‘ 𝑀 ) ) |
| 64 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) → ( 𝐸 ∪ 𝐹 ) ⊆ ( Base ‘ 𝑀 ) ) |
| 65 |
64
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) → 𝑥 ∈ ( Base ‘ 𝑀 ) ) |
| 66 |
65
|
ad3antrrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → 𝑥 ∈ ( Base ‘ 𝑀 ) ) |
| 67 |
39 1
|
gsumccatsn |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑤 ∈ Word ( Base ‘ 𝑀 ) ∧ 𝑥 ∈ ( Base ‘ 𝑀 ) ) → ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( ( 𝑀 Σg 𝑤 ) + 𝑥 ) ) |
| 68 |
54 63 66 67
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( ( 𝑀 Σg 𝑤 ) + 𝑥 ) ) |
| 69 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) |
| 70 |
69
|
oveq1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( ( 𝑀 Σg 𝑤 ) + 𝑥 ) = ( ( 𝑒 + 𝑓 ) + 𝑥 ) ) |
| 71 |
56
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) → 𝐸 ⊆ ( Base ‘ 𝑀 ) ) |
| 72 |
71
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) → 𝑒 ∈ ( Base ‘ 𝑀 ) ) |
| 73 |
72
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → 𝑒 ∈ ( Base ‘ 𝑀 ) ) |
| 74 |
58
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) → 𝐹 ⊆ ( Base ‘ 𝑀 ) ) |
| 75 |
74
|
sselda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) → 𝑓 ∈ ( Base ‘ 𝑀 ) ) |
| 76 |
75
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → 𝑓 ∈ ( Base ‘ 𝑀 ) ) |
| 77 |
2
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → 𝑀 ∈ CMnd ) |
| 78 |
39 1
|
cmncom |
⊢ ( ( 𝑀 ∈ CMnd ∧ 𝑓 ∈ ( Base ‘ 𝑀 ) ∧ 𝑥 ∈ ( Base ‘ 𝑀 ) ) → ( 𝑓 + 𝑥 ) = ( 𝑥 + 𝑓 ) ) |
| 79 |
77 76 66 78
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( 𝑓 + 𝑥 ) = ( 𝑥 + 𝑓 ) ) |
| 80 |
39 1 54 73 76 66 79
|
mnd32g |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( ( 𝑒 + 𝑓 ) + 𝑥 ) = ( ( 𝑒 + 𝑥 ) + 𝑓 ) ) |
| 81 |
68 70 80
|
3eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( ( 𝑒 + 𝑥 ) + 𝑓 ) ) |
| 82 |
81
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐸 ) → ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( ( 𝑒 + 𝑥 ) + 𝑓 ) ) |
| 83 |
46 48 52 53 82
|
2rspcedvdw |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐸 ) → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) |
| 84 |
|
oveq1 |
⊢ ( 𝑖 = 𝑒 → ( 𝑖 + 𝑗 ) = ( 𝑒 + 𝑗 ) ) |
| 85 |
84
|
eqeq2d |
⊢ ( 𝑖 = 𝑒 → ( ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ↔ ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑒 + 𝑗 ) ) ) |
| 86 |
|
oveq2 |
⊢ ( 𝑗 = ( 𝑓 + 𝑥 ) → ( 𝑒 + 𝑗 ) = ( 𝑒 + ( 𝑓 + 𝑥 ) ) ) |
| 87 |
86
|
eqeq2d |
⊢ ( 𝑗 = ( 𝑓 + 𝑥 ) → ( ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑒 + 𝑗 ) ↔ ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑒 + ( 𝑓 + 𝑥 ) ) ) ) |
| 88 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐹 ) → 𝑒 ∈ 𝐸 ) |
| 89 |
4
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐹 ) → 𝐹 ∈ ( SubMnd ‘ 𝑀 ) ) |
| 90 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐹 ) → 𝑓 ∈ 𝐹 ) |
| 91 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐹 ) → 𝑥 ∈ 𝐹 ) |
| 92 |
1 89 90 91
|
submcld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐹 ) → ( 𝑓 + 𝑥 ) ∈ 𝐹 ) |
| 93 |
39 1 54 73 76 66
|
mndassd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( ( 𝑒 + 𝑓 ) + 𝑥 ) = ( 𝑒 + ( 𝑓 + 𝑥 ) ) ) |
| 94 |
68 70 93
|
3eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑒 + ( 𝑓 + 𝑥 ) ) ) |
| 95 |
94
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐹 ) → ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑒 + ( 𝑓 + 𝑥 ) ) ) |
| 96 |
85 87 88 92 95
|
2rspcedvdw |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) ∧ 𝑥 ∈ 𝐹 ) → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) |
| 97 |
|
elun |
⊢ ( 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ↔ ( 𝑥 ∈ 𝐸 ∨ 𝑥 ∈ 𝐹 ) ) |
| 98 |
97
|
biimpi |
⊢ ( 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) → ( 𝑥 ∈ 𝐸 ∨ 𝑥 ∈ 𝐹 ) ) |
| 99 |
98
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( 𝑥 ∈ 𝐸 ∨ 𝑥 ∈ 𝐹 ) ) |
| 100 |
83 96 99
|
mpjaodan |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑒 ∈ 𝐸 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) |
| 101 |
100
|
r19.29ffa |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) ∧ ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) |
| 102 |
101
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) → ( ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) ) |
| 103 |
102
|
expl |
⊢ ( 𝜑 → ( ( 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) → ( ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) ) ) |
| 104 |
103
|
com12 |
⊢ ( ( 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) → ( 𝜑 → ( ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) ) ) |
| 105 |
104
|
a2d |
⊢ ( ( 𝑤 ∈ Word ( 𝐸 ∪ 𝐹 ) ∧ 𝑥 ∈ ( 𝐸 ∪ 𝐹 ) ) → ( ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑤 ) = ( 𝑒 + 𝑓 ) ) → ( 𝜑 → ∃ 𝑖 ∈ 𝐸 ∃ 𝑗 ∈ 𝐹 ( 𝑀 Σg ( 𝑤 ++ 〈“ 𝑥 ”〉 ) ) = ( 𝑖 + 𝑗 ) ) ) ) |
| 106 |
9 13 23 27 44 105
|
wrdind |
⊢ ( 𝑊 ∈ Word ( 𝐸 ∪ 𝐹 ) → ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑊 ) = ( 𝑒 + 𝑓 ) ) ) |
| 107 |
5 106
|
mpcom |
⊢ ( 𝜑 → ∃ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 ( 𝑀 Σg 𝑊 ) = ( 𝑒 + 𝑓 ) ) |