| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mhpmulcl.h |
⊢ 𝐻 = ( 𝐼 mHomP 𝑅 ) |
| 2 |
|
mhpmulcl.y |
⊢ 𝑌 = ( 𝐼 mPoly 𝑅 ) |
| 3 |
|
mhpmulcl.t |
⊢ · = ( .r ‘ 𝑌 ) |
| 4 |
|
mhpmulcl.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
| 5 |
|
mhpmulcl.p |
⊢ ( 𝜑 → 𝑃 ∈ ( 𝐻 ‘ 𝑀 ) ) |
| 6 |
|
mhpmulcl.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝐻 ‘ 𝑁 ) ) |
| 7 |
|
breq2 |
⊢ ( 𝑑 = 𝑥 → ( 𝑐 ∘r ≤ 𝑑 ↔ 𝑐 ∘r ≤ 𝑥 ) ) |
| 8 |
7
|
rabbidv |
⊢ ( 𝑑 = 𝑥 → { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑑 } = { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) |
| 9 |
|
fvoveq1 |
⊢ ( 𝑑 = 𝑥 → ( 𝑄 ‘ ( 𝑑 ∘f − 𝑒 ) ) = ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) |
| 10 |
9
|
oveq2d |
⊢ ( 𝑑 = 𝑥 → ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑑 ∘f − 𝑒 ) ) ) = ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) |
| 11 |
8 10
|
mpteq12dv |
⊢ ( 𝑑 = 𝑥 → ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑑 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑑 ∘f − 𝑒 ) ) ) ) = ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) |
| 12 |
11
|
oveq2d |
⊢ ( 𝑑 = 𝑥 → ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑑 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑑 ∘f − 𝑒 ) ) ) ) ) = ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) ) |
| 13 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
| 14 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
| 15 |
|
eqid |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
| 16 |
1 2 13 5
|
mhpmpl |
⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝑌 ) ) |
| 17 |
1 2 13 6
|
mhpmpl |
⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝑌 ) ) |
| 18 |
2 13 14 3 15 16 17
|
mplmul |
⊢ ( 𝜑 → ( 𝑃 · 𝑄 ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑑 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑑 ∘f − 𝑒 ) ) ) ) ) ) ) |
| 19 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑃 · 𝑄 ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑑 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑑 ∘f − 𝑒 ) ) ) ) ) ) ) |
| 20 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
| 21 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) ∈ V ) |
| 22 |
12 19 20 21
|
fvmptd4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑃 · 𝑄 ) ‘ 𝑥 ) = ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) ) |
| 23 |
22
|
neeq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑃 · 𝑄 ) ‘ 𝑥 ) ≠ ( 0g ‘ 𝑅 ) ↔ ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) ≠ ( 0g ‘ 𝑅 ) ) ) |
| 24 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → 𝜑 ) |
| 25 |
|
oveq2 |
⊢ ( 𝑐 = 𝑒 → ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ) |
| 26 |
25
|
eqeq1d |
⊢ ( 𝑐 = 𝑒 → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑀 ↔ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) = 𝑀 ) ) |
| 27 |
26
|
necon3bbid |
⊢ ( 𝑐 = 𝑒 → ( ¬ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑀 ↔ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) ) |
| 28 |
|
elrabi |
⊢ ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } → 𝑒 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
| 29 |
28
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → 𝑒 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
| 30 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) |
| 31 |
27 29 30
|
elrabd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ¬ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑀 } ) |
| 32 |
|
notrab |
⊢ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑀 } ) = { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ¬ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑀 } |
| 33 |
31 32
|
eleqtrrdi |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → 𝑒 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑀 } ) ) |
| 34 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
| 35 |
2 34 13 15 16
|
mplelf |
⊢ ( 𝜑 → 𝑃 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
| 36 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
| 37 |
1 36 15 5
|
mhpdeg |
⊢ ( 𝜑 → ( 𝑃 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑀 } ) |
| 38 |
|
fvexd |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ V ) |
| 39 |
35 37 5 38
|
suppssrg |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑀 } ) ) → ( 𝑃 ‘ 𝑒 ) = ( 0g ‘ 𝑅 ) ) |
| 40 |
24 33 39
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → ( 𝑃 ‘ 𝑒 ) = ( 0g ‘ 𝑅 ) ) |
| 41 |
40
|
oveq1d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) = ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) |
| 42 |
4
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → 𝑅 ∈ Ring ) |
| 43 |
17
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → 𝑄 ∈ ( Base ‘ 𝑌 ) ) |
| 44 |
2 34 13 15 43
|
mplelf |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → 𝑄 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
| 45 |
|
eqid |
⊢ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } = { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } |
| 46 |
15 45
|
psrbagconcl |
⊢ ( ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑥 ∘f − 𝑒 ) ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) |
| 47 |
46
|
ad5ant24 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → ( 𝑥 ∘f − 𝑒 ) ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) |
| 48 |
|
elrabi |
⊢ ( ( 𝑥 ∘f − 𝑒 ) ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } → ( 𝑥 ∘f − 𝑒 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
| 49 |
47 48
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → ( 𝑥 ∘f − 𝑒 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
| 50 |
44 49
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ∈ ( Base ‘ 𝑅 ) ) |
| 51 |
34 14 36 42 50
|
ringlzd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) = ( 0g ‘ 𝑅 ) ) |
| 52 |
41 51
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ) → ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) = ( 0g ‘ 𝑅 ) ) |
| 53 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → 𝜑 ) |
| 54 |
|
oveq2 |
⊢ ( 𝑐 = ( 𝑥 ∘f − 𝑒 ) → ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ) |
| 55 |
54
|
eqeq1d |
⊢ ( 𝑐 = ( 𝑥 ∘f − 𝑒 ) → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑁 ↔ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) = 𝑁 ) ) |
| 56 |
55
|
necon3bbid |
⊢ ( 𝑐 = ( 𝑥 ∘f − 𝑒 ) → ( ¬ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑁 ↔ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) ) |
| 57 |
46
|
ad5ant24 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( 𝑥 ∘f − 𝑒 ) ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) |
| 58 |
57 48
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( 𝑥 ∘f − 𝑒 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
| 59 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) |
| 60 |
56 58 59
|
elrabd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( 𝑥 ∘f − 𝑒 ) ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ¬ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑁 } ) |
| 61 |
|
notrab |
⊢ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑁 } ) = { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ¬ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑁 } |
| 62 |
60 61
|
eleqtrrdi |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( 𝑥 ∘f − 𝑒 ) ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑁 } ) ) |
| 63 |
2 34 13 15 17
|
mplelf |
⊢ ( 𝜑 → 𝑄 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
| 64 |
1 36 15 6
|
mhpdeg |
⊢ ( 𝜑 → ( 𝑄 supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑁 } ) |
| 65 |
63 64 6 38
|
suppssrg |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∘f − 𝑒 ) ∈ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∖ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( ( ℂfld ↾s ℕ0 ) Σg 𝑐 ) = 𝑁 } ) ) → ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) = ( 0g ‘ 𝑅 ) ) |
| 66 |
53 62 65
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) = ( 0g ‘ 𝑅 ) ) |
| 67 |
66
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) = ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) ) |
| 68 |
4
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → 𝑅 ∈ Ring ) |
| 69 |
16
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → 𝑃 ∈ ( Base ‘ 𝑌 ) ) |
| 70 |
2 34 13 15 69
|
mplelf |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → 𝑃 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
| 71 |
28
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → 𝑒 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
| 72 |
70 71
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( 𝑃 ‘ 𝑒 ) ∈ ( Base ‘ 𝑅 ) ) |
| 73 |
34 14 36 68 72
|
ringrzd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
| 74 |
67 73
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) → ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) = ( 0g ‘ 𝑅 ) ) |
| 75 |
|
nn0subm |
⊢ ℕ0 ∈ ( SubMnd ‘ ℂfld ) |
| 76 |
|
eqid |
⊢ ( ℂfld ↾s ℕ0 ) = ( ℂfld ↾s ℕ0 ) |
| 77 |
76
|
submbas |
⊢ ( ℕ0 ∈ ( SubMnd ‘ ℂfld ) → ℕ0 = ( Base ‘ ( ℂfld ↾s ℕ0 ) ) ) |
| 78 |
75 77
|
ax-mp |
⊢ ℕ0 = ( Base ‘ ( ℂfld ↾s ℕ0 ) ) |
| 79 |
|
cnfld0 |
⊢ 0 = ( 0g ‘ ℂfld ) |
| 80 |
76 79
|
subm0 |
⊢ ( ℕ0 ∈ ( SubMnd ‘ ℂfld ) → 0 = ( 0g ‘ ( ℂfld ↾s ℕ0 ) ) ) |
| 81 |
75 80
|
ax-mp |
⊢ 0 = ( 0g ‘ ( ℂfld ↾s ℕ0 ) ) |
| 82 |
|
nn0ex |
⊢ ℕ0 ∈ V |
| 83 |
|
cnfldadd |
⊢ + = ( +g ‘ ℂfld ) |
| 84 |
76 83
|
ressplusg |
⊢ ( ℕ0 ∈ V → + = ( +g ‘ ( ℂfld ↾s ℕ0 ) ) ) |
| 85 |
82 84
|
ax-mp |
⊢ + = ( +g ‘ ( ℂfld ↾s ℕ0 ) ) |
| 86 |
|
cnring |
⊢ ℂfld ∈ Ring |
| 87 |
|
ringcmn |
⊢ ( ℂfld ∈ Ring → ℂfld ∈ CMnd ) |
| 88 |
86 87
|
ax-mp |
⊢ ℂfld ∈ CMnd |
| 89 |
76
|
submcmn |
⊢ ( ( ℂfld ∈ CMnd ∧ ℕ0 ∈ ( SubMnd ‘ ℂfld ) ) → ( ℂfld ↾s ℕ0 ) ∈ CMnd ) |
| 90 |
88 75 89
|
mp2an |
⊢ ( ℂfld ↾s ℕ0 ) ∈ CMnd |
| 91 |
90
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ℂfld ↾s ℕ0 ) ∈ CMnd ) |
| 92 |
|
reldmmhp |
⊢ Rel dom mHomP |
| 93 |
92 1 5
|
elfvov1 |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
| 94 |
93
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝐼 ∈ V ) |
| 95 |
28
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑒 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) |
| 96 |
15
|
psrbagf |
⊢ ( 𝑒 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑒 : 𝐼 ⟶ ℕ0 ) |
| 97 |
95 96
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑒 : 𝐼 ⟶ ℕ0 ) |
| 98 |
15
|
psrbagf |
⊢ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑥 : 𝐼 ⟶ ℕ0 ) |
| 99 |
98
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑥 : 𝐼 ⟶ ℕ0 ) |
| 100 |
99
|
ffnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑥 Fn 𝐼 ) |
| 101 |
97
|
ffnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑒 Fn 𝐼 ) |
| 102 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
| 103 |
|
eqidd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑥 ‘ 𝑖 ) = ( 𝑥 ‘ 𝑖 ) ) |
| 104 |
|
eqidd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑒 ‘ 𝑖 ) = ( 𝑒 ‘ 𝑖 ) ) |
| 105 |
100 101 94 94 102 103 104
|
offval |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑥 ∘f − 𝑒 ) = ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑥 ‘ 𝑖 ) − ( 𝑒 ‘ 𝑖 ) ) ) ) |
| 106 |
|
simpl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ) |
| 107 |
|
breq1 |
⊢ ( 𝑐 = 𝑒 → ( 𝑐 ∘r ≤ 𝑥 ↔ 𝑒 ∘r ≤ 𝑥 ) ) |
| 108 |
107
|
elrab |
⊢ ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↔ ( 𝑒 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑒 ∘r ≤ 𝑥 ) ) |
| 109 |
108
|
simprbi |
⊢ ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } → 𝑒 ∘r ≤ 𝑥 ) |
| 110 |
109
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → 𝑒 ∘r ≤ 𝑥 ) |
| 111 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → 𝑖 ∈ 𝐼 ) |
| 112 |
101 100 94 94 102 104 103
|
ofrval |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑒 ∘r ≤ 𝑥 ∧ 𝑖 ∈ 𝐼 ) → ( 𝑒 ‘ 𝑖 ) ≤ ( 𝑥 ‘ 𝑖 ) ) |
| 113 |
106 110 111 112
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑒 ‘ 𝑖 ) ≤ ( 𝑥 ‘ 𝑖 ) ) |
| 114 |
97
|
ffvelcdmda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑒 ‘ 𝑖 ) ∈ ℕ0 ) |
| 115 |
99
|
ffvelcdmda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑥 ‘ 𝑖 ) ∈ ℕ0 ) |
| 116 |
|
nn0sub |
⊢ ( ( ( 𝑒 ‘ 𝑖 ) ∈ ℕ0 ∧ ( 𝑥 ‘ 𝑖 ) ∈ ℕ0 ) → ( ( 𝑒 ‘ 𝑖 ) ≤ ( 𝑥 ‘ 𝑖 ) ↔ ( ( 𝑥 ‘ 𝑖 ) − ( 𝑒 ‘ 𝑖 ) ) ∈ ℕ0 ) ) |
| 117 |
114 115 116
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑒 ‘ 𝑖 ) ≤ ( 𝑥 ‘ 𝑖 ) ↔ ( ( 𝑥 ‘ 𝑖 ) − ( 𝑒 ‘ 𝑖 ) ) ∈ ℕ0 ) ) |
| 118 |
113 117
|
mpbid |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑥 ‘ 𝑖 ) − ( 𝑒 ‘ 𝑖 ) ) ∈ ℕ0 ) |
| 119 |
105 118
|
fmpt3d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑥 ∘f − 𝑒 ) : 𝐼 ⟶ ℕ0 ) |
| 120 |
97
|
ffund |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → Fun 𝑒 ) |
| 121 |
|
c0ex |
⊢ 0 ∈ V |
| 122 |
94 121
|
jctir |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝐼 ∈ V ∧ 0 ∈ V ) ) |
| 123 |
|
fsuppeq |
⊢ ( ( 𝐼 ∈ V ∧ 0 ∈ V ) → ( 𝑒 : 𝐼 ⟶ ℕ0 → ( 𝑒 supp 0 ) = ( ◡ 𝑒 “ ( ℕ0 ∖ { 0 } ) ) ) ) |
| 124 |
122 97 123
|
sylc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑒 supp 0 ) = ( ◡ 𝑒 “ ( ℕ0 ∖ { 0 } ) ) ) |
| 125 |
|
dfn2 |
⊢ ℕ = ( ℕ0 ∖ { 0 } ) |
| 126 |
125
|
imaeq2i |
⊢ ( ◡ 𝑒 “ ℕ ) = ( ◡ 𝑒 “ ( ℕ0 ∖ { 0 } ) ) |
| 127 |
124 126
|
eqtr4di |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑒 supp 0 ) = ( ◡ 𝑒 “ ℕ ) ) |
| 128 |
15
|
psrbag |
⊢ ( 𝐼 ∈ V → ( 𝑒 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↔ ( 𝑒 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝑒 “ ℕ ) ∈ Fin ) ) ) |
| 129 |
94 128
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑒 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↔ ( 𝑒 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝑒 “ ℕ ) ∈ Fin ) ) ) |
| 130 |
95 129
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑒 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝑒 “ ℕ ) ∈ Fin ) ) |
| 131 |
130
|
simprd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ◡ 𝑒 “ ℕ ) ∈ Fin ) |
| 132 |
127 131
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑒 supp 0 ) ∈ Fin ) |
| 133 |
95
|
elexd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑒 ∈ V ) |
| 134 |
|
isfsupp |
⊢ ( ( 𝑒 ∈ V ∧ 0 ∈ V ) → ( 𝑒 finSupp 0 ↔ ( Fun 𝑒 ∧ ( 𝑒 supp 0 ) ∈ Fin ) ) ) |
| 135 |
133 121 134
|
sylancl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑒 finSupp 0 ↔ ( Fun 𝑒 ∧ ( 𝑒 supp 0 ) ∈ Fin ) ) ) |
| 136 |
120 132 135
|
mpbir2and |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑒 finSupp 0 ) |
| 137 |
|
ovexd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑥 ∘f − 𝑒 ) ∈ V ) |
| 138 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
| 139 |
138
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 0 ∈ ℕ0 ) |
| 140 |
100 101 94 94
|
offun |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → Fun ( 𝑥 ∘f − 𝑒 ) ) |
| 141 |
15
|
psrbagfsupp |
⊢ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑥 finSupp 0 ) |
| 142 |
141
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑥 finSupp 0 ) |
| 143 |
142 136
|
fsuppunfi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( 𝑥 supp 0 ) ∪ ( 𝑒 supp 0 ) ) ∈ Fin ) |
| 144 |
|
0m0e0 |
⊢ ( 0 − 0 ) = 0 |
| 145 |
144
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 0 − 0 ) = 0 ) |
| 146 |
94 139 99 97 145
|
suppofssd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( 𝑥 ∘f − 𝑒 ) supp 0 ) ⊆ ( ( 𝑥 supp 0 ) ∪ ( 𝑒 supp 0 ) ) ) |
| 147 |
143 146
|
ssfid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( 𝑥 ∘f − 𝑒 ) supp 0 ) ∈ Fin ) |
| 148 |
137 139 140 147
|
isfsuppd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑥 ∘f − 𝑒 ) finSupp 0 ) |
| 149 |
78 81 85 91 94 97 119 136 148
|
gsumadd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑒 ∘f + ( 𝑥 ∘f − 𝑒 ) ) ) = ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) + ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ) ) |
| 150 |
97
|
ffvelcdmda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑏 ∈ 𝐼 ) → ( 𝑒 ‘ 𝑏 ) ∈ ℕ0 ) |
| 151 |
150
|
nn0cnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑏 ∈ 𝐼 ) → ( 𝑒 ‘ 𝑏 ) ∈ ℂ ) |
| 152 |
99
|
ffvelcdmda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑏 ∈ 𝐼 ) → ( 𝑥 ‘ 𝑏 ) ∈ ℕ0 ) |
| 153 |
152
|
nn0cnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑏 ∈ 𝐼 ) → ( 𝑥 ‘ 𝑏 ) ∈ ℂ ) |
| 154 |
151 153
|
pncan3d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑏 ∈ 𝐼 ) → ( ( 𝑒 ‘ 𝑏 ) + ( ( 𝑥 ‘ 𝑏 ) − ( 𝑒 ‘ 𝑏 ) ) ) = ( 𝑥 ‘ 𝑏 ) ) |
| 155 |
154
|
mpteq2dva |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑏 ∈ 𝐼 ↦ ( ( 𝑒 ‘ 𝑏 ) + ( ( 𝑥 ‘ 𝑏 ) − ( 𝑒 ‘ 𝑏 ) ) ) ) = ( 𝑏 ∈ 𝐼 ↦ ( 𝑥 ‘ 𝑏 ) ) ) |
| 156 |
|
fvexd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑏 ∈ 𝐼 ) → ( 𝑒 ‘ 𝑏 ) ∈ V ) |
| 157 |
|
ovexd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) ∧ 𝑏 ∈ 𝐼 ) → ( ( 𝑥 ‘ 𝑏 ) − ( 𝑒 ‘ 𝑏 ) ) ∈ V ) |
| 158 |
97
|
feqmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑒 = ( 𝑏 ∈ 𝐼 ↦ ( 𝑒 ‘ 𝑏 ) ) ) |
| 159 |
99
|
feqmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → 𝑥 = ( 𝑏 ∈ 𝐼 ↦ ( 𝑥 ‘ 𝑏 ) ) ) |
| 160 |
94 152 150 159 158
|
offval2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑥 ∘f − 𝑒 ) = ( 𝑏 ∈ 𝐼 ↦ ( ( 𝑥 ‘ 𝑏 ) − ( 𝑒 ‘ 𝑏 ) ) ) ) |
| 161 |
94 156 157 158 160
|
offval2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑒 ∘f + ( 𝑥 ∘f − 𝑒 ) ) = ( 𝑏 ∈ 𝐼 ↦ ( ( 𝑒 ‘ 𝑏 ) + ( ( 𝑥 ‘ 𝑏 ) − ( 𝑒 ‘ 𝑏 ) ) ) ) ) |
| 162 |
155 161 159
|
3eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( 𝑒 ∘f + ( 𝑥 ∘f − 𝑒 ) ) = 𝑥 ) |
| 163 |
162
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑒 ∘f + ( 𝑥 ∘f − 𝑒 ) ) ) = ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ) |
| 164 |
149 163
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) + ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ) = ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ) |
| 165 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) |
| 166 |
164 165
|
eqnetrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) + ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ) ≠ ( 𝑀 + 𝑁 ) ) |
| 167 |
|
oveq12 |
⊢ ( ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) = 𝑀 ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) = 𝑁 ) → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) + ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ) = ( 𝑀 + 𝑁 ) ) |
| 168 |
167
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) = 𝑀 ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) = 𝑁 ) → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) + ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ) = ( 𝑀 + 𝑁 ) ) ) |
| 169 |
168
|
necon3ad |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) + ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ) ≠ ( 𝑀 + 𝑁 ) → ¬ ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) = 𝑀 ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) = 𝑁 ) ) ) |
| 170 |
166 169
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ¬ ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) = 𝑀 ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) = 𝑁 ) ) |
| 171 |
|
neorian |
⊢ ( ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ∨ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) ↔ ¬ ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) = 𝑀 ∧ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) = 𝑁 ) ) |
| 172 |
170 171
|
sylibr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑒 ) ≠ 𝑀 ∨ ( ( ℂfld ↾s ℕ0 ) Σg ( 𝑥 ∘f − 𝑒 ) ) ≠ 𝑁 ) ) |
| 173 |
52 74 172
|
mpjaodan |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) ∧ 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ) → ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) = ( 0g ‘ 𝑅 ) ) |
| 174 |
173
|
mpteq2dva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) → ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) = ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( 0g ‘ 𝑅 ) ) ) |
| 175 |
174
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) → ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) = ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( 0g ‘ 𝑅 ) ) ) ) |
| 176 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
| 177 |
4 176
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
| 178 |
177
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) → 𝑅 ∈ Mnd ) |
| 179 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
| 180 |
179
|
rabex |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V |
| 181 |
180
|
rabex |
⊢ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ∈ V |
| 182 |
36
|
gsumz |
⊢ ( ( 𝑅 ∈ Mnd ∧ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ∈ V ) → ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( 0g ‘ 𝑅 ) ) ) = ( 0g ‘ 𝑅 ) ) |
| 183 |
178 181 182
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) → ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( 0g ‘ 𝑅 ) ) ) = ( 0g ‘ 𝑅 ) ) |
| 184 |
175 183
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) ) → ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
| 185 |
184
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) ≠ ( 𝑀 + 𝑁 ) → ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) = ( 0g ‘ 𝑅 ) ) ) |
| 186 |
185
|
necon1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑅 Σg ( 𝑒 ∈ { 𝑐 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑐 ∘r ≤ 𝑥 } ↦ ( ( 𝑃 ‘ 𝑒 ) ( .r ‘ 𝑅 ) ( 𝑄 ‘ ( 𝑥 ∘f − 𝑒 ) ) ) ) ) ≠ ( 0g ‘ 𝑅 ) → ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) = ( 𝑀 + 𝑁 ) ) ) |
| 187 |
23 186
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑃 · 𝑄 ) ‘ 𝑥 ) ≠ ( 0g ‘ 𝑅 ) → ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) = ( 𝑀 + 𝑁 ) ) ) |
| 188 |
187
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ( ( ( 𝑃 · 𝑄 ) ‘ 𝑥 ) ≠ ( 0g ‘ 𝑅 ) → ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) = ( 𝑀 + 𝑁 ) ) ) |
| 189 |
1 5
|
mhprcl |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
| 190 |
1 6
|
mhprcl |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
| 191 |
189 190
|
nn0addcld |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ℕ0 ) |
| 192 |
2 93 4
|
mplringd |
⊢ ( 𝜑 → 𝑌 ∈ Ring ) |
| 193 |
13 3 192 16 17
|
ringcld |
⊢ ( 𝜑 → ( 𝑃 · 𝑄 ) ∈ ( Base ‘ 𝑌 ) ) |
| 194 |
1 2 13 36 15 191 193
|
ismhp3 |
⊢ ( 𝜑 → ( ( 𝑃 · 𝑄 ) ∈ ( 𝐻 ‘ ( 𝑀 + 𝑁 ) ) ↔ ∀ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ( ( ( 𝑃 · 𝑄 ) ‘ 𝑥 ) ≠ ( 0g ‘ 𝑅 ) → ( ( ℂfld ↾s ℕ0 ) Σg 𝑥 ) = ( 𝑀 + 𝑁 ) ) ) ) |
| 195 |
188 194
|
mpbird |
⊢ ( 𝜑 → ( 𝑃 · 𝑄 ) ∈ ( 𝐻 ‘ ( 𝑀 + 𝑁 ) ) ) |