Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fmuldfeq Structured version   Visualization version   GIF version

Theorem fmuldfeq 38650
Description: X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
fmuldfeq.1 𝑖𝜑
fmuldfeq.2 𝑡𝑌
fmuldfeq.3 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
fmuldfeq.4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
fmuldfeq.5 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
fmuldfeq.6 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
fmuldfeq.7 (𝜑𝑇 ∈ V)
fmuldfeq.8 (𝜑𝑀 ∈ ℕ)
fmuldfeq.9 (𝜑𝑈:(1...𝑀)⟶𝑌)
fmuldfeq.10 ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)
fmuldfeq.11 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
Assertion
Ref Expression
fmuldfeq ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
Distinct variable groups:   𝑡,𝑇   𝑓,𝑔,𝑡,𝑇   𝑓,𝑖,𝑡,𝑇   𝑓,𝐹,𝑔   𝑓,𝑀,𝑔   𝑈,𝑓,𝑔,𝑡   𝑓,𝑌,𝑔   𝜑,𝑓,𝑔   𝑖,𝑀   𝑈,𝑖
Allowed substitution hints:   𝜑(𝑡,𝑖)   𝑃(𝑡,𝑓,𝑔,𝑖)   𝐹(𝑡,𝑖)   𝑀(𝑡)   𝑋(𝑡,𝑓,𝑔,𝑖)   𝑌(𝑡,𝑖)   𝑍(𝑡,𝑓,𝑔,𝑖)

Proof of Theorem fmuldfeq
Dummy variables 𝑘 𝑏 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmuldfeq.8 . . . . . 6 (𝜑𝑀 ∈ ℕ)
21nnge1d 10940 . . . . 5 (𝜑 → 1 ≤ 𝑀)
32adantr 480 . . . 4 ((𝜑𝑡𝑇) → 1 ≤ 𝑀)
4 nnre 10904 . . . . . 6 (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
5 leid 10012 . . . . . 6 (𝑀 ∈ ℝ → 𝑀𝑀)
61, 4, 53syl 18 . . . . 5 (𝜑𝑀𝑀)
76adantr 480 . . . 4 ((𝜑𝑡𝑇) → 𝑀𝑀)
81nnzd 11357 . . . . . 6 (𝜑𝑀 ∈ ℤ)
98adantr 480 . . . . 5 ((𝜑𝑡𝑇) → 𝑀 ∈ ℤ)
10 1zzd 11285 . . . . 5 ((𝜑𝑡𝑇) → 1 ∈ ℤ)
11 elfz 12203 . . . . 5 ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (1...𝑀) ↔ (1 ≤ 𝑀𝑀𝑀)))
129, 10, 9, 11syl3anc 1318 . . . 4 ((𝜑𝑡𝑇) → (𝑀 ∈ (1...𝑀) ↔ (1 ≤ 𝑀𝑀𝑀)))
133, 7, 12mpbir2and 959 . . 3 ((𝜑𝑡𝑇) → 𝑀 ∈ (1...𝑀))
1413ad2ant1 1075 . . . 4 ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → 𝑀 ∈ ℕ)
15 eleq1 2676 . . . . . . 7 (𝑚 = 1 → (𝑚 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
16153anbi3d 1397 . . . . . 6 (𝑚 = 1 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀))))
17 fveq2 6103 . . . . . . . 8 (𝑚 = 1 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘1))
1817fveq1d 6105 . . . . . . 7 (𝑚 = 1 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
19 fveq2 6103 . . . . . . 7 (𝑚 = 1 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘1))
2018, 19eqeq12d 2625 . . . . . 6 (𝑚 = 1 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1)))
2116, 20imbi12d 333 . . . . 5 (𝑚 = 1 → (((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚)) ↔ ((𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))))
22 eleq1 2676 . . . . . . 7 (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑀) ↔ 𝑛 ∈ (1...𝑀)))
23223anbi3d 1397 . . . . . 6 (𝑚 = 𝑛 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇𝑛 ∈ (1...𝑀))))
24 fveq2 6103 . . . . . . . 8 (𝑚 = 𝑛 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑛))
2524fveq1d 6105 . . . . . . 7 (𝑚 = 𝑛 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡))
26 fveq2 6103 . . . . . . 7 (𝑚 = 𝑛 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘𝑛))
2725, 26eqeq12d 2625 . . . . . 6 (𝑚 = 𝑛 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
2823, 27imbi12d 333 . . . . 5 (𝑚 = 𝑛 → (((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚)) ↔ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))))
29 eleq1 2676 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑚 ∈ (1...𝑀) ↔ (𝑛 + 1) ∈ (1...𝑀)))
30293anbi3d 1397 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))))
31 fveq2 6103 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘(𝑛 + 1)))
3231fveq1d 6105 . . . . . . 7 (𝑚 = (𝑛 + 1) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡))
33 fveq2 6103 . . . . . . 7 (𝑚 = (𝑛 + 1) → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1)))
3432, 33eqeq12d 2625 . . . . . 6 (𝑚 = (𝑛 + 1) → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1))))
3530, 34imbi12d 333 . . . . 5 (𝑚 = (𝑛 + 1) → (((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚)) ↔ ((𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1)))))
36 eleq1 2676 . . . . . . 7 (𝑚 = 𝑀 → (𝑚 ∈ (1...𝑀) ↔ 𝑀 ∈ (1...𝑀)))
37363anbi3d 1397 . . . . . 6 (𝑚 = 𝑀 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇𝑀 ∈ (1...𝑀))))
38 fveq2 6103 . . . . . . . 8 (𝑚 = 𝑀 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑀))
3938fveq1d 6105 . . . . . . 7 (𝑚 = 𝑀 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
40 fveq2 6103 . . . . . . 7 (𝑚 = 𝑀 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘𝑀))
4139, 40eqeq12d 2625 . . . . . 6 (𝑚 = 𝑀 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀)))
4237, 41imbi12d 333 . . . . 5 (𝑚 = 𝑀 → (((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚)) ↔ ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))))
43 1z 11284 . . . . . . . 8 1 ∈ ℤ
44 seq1 12676 . . . . . . . 8 (1 ∈ ℤ → (seq1( · , (𝐹𝑡))‘1) = ((𝐹𝑡)‘1))
4543, 44ax-mp 5 . . . . . . 7 (seq1( · , (𝐹𝑡))‘1) = ((𝐹𝑡)‘1)
46 1le1 10534 . . . . . . . . . . . . 13 1 ≤ 1
4746a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ≤ 1)
48 1zzd 11285 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ ℤ)
49 elfz 12203 . . . . . . . . . . . . 13 ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 ∈ (1...𝑀) ↔ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
5048, 48, 8, 49syl3anc 1318 . . . . . . . . . . . 12 (𝜑 → (1 ∈ (1...𝑀) ↔ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
5147, 2, 50mpbir2and 959 . . . . . . . . . . 11 (𝜑 → 1 ∈ (1...𝑀))
52 nfv 1830 . . . . . . . . . . . . 13 𝑖 𝑡𝑇
53 fmuldfeq.5 . . . . . . . . . . . . . . . . 17 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
54 nfcv 2751 . . . . . . . . . . . . . . . . . 18 𝑖𝑇
55 nfmpt1 4675 . . . . . . . . . . . . . . . . . 18 𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
5654, 55nfmpt 4674 . . . . . . . . . . . . . . . . 17 𝑖(𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
5753, 56nfcxfr 2749 . . . . . . . . . . . . . . . 16 𝑖𝐹
58 nfcv 2751 . . . . . . . . . . . . . . . 16 𝑖𝑡
5957, 58nffv 6110 . . . . . . . . . . . . . . 15 𝑖(𝐹𝑡)
60 nfcv 2751 . . . . . . . . . . . . . . 15 𝑖1
6159, 60nffv 6110 . . . . . . . . . . . . . 14 𝑖((𝐹𝑡)‘1)
62 nffvmpt1 6111 . . . . . . . . . . . . . 14 𝑖((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)
6361, 62nfeq 2762 . . . . . . . . . . . . 13 𝑖((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)
6452, 63nfim 1813 . . . . . . . . . . . 12 𝑖(𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
65 fveq2 6103 . . . . . . . . . . . . . 14 (𝑖 = 1 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘1))
66 fveq2 6103 . . . . . . . . . . . . . 14 (𝑖 = 1 → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
6765, 66eqeq12d 2625 . . . . . . . . . . . . 13 (𝑖 = 1 → (((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) ↔ ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)))
6867imbi2d 329 . . . . . . . . . . . 12 (𝑖 = 1 → ((𝑡𝑇 → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖)) ↔ (𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))))
69 ovex 6577 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ V
7069mptex 6390 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V
7153fvmpt2 6200 . . . . . . . . . . . . . 14 ((𝑡𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V) → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7270, 71mpan2 703 . . . . . . . . . . . . 13 (𝑡𝑇 → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7372fveq1d 6105 . . . . . . . . . . . 12 (𝑡𝑇 → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
7464, 68, 73vtoclg1f 3238 . . . . . . . . . . 11 (1 ∈ (1...𝑀) → (𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)))
7551, 74syl 17 . . . . . . . . . 10 (𝜑 → (𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)))
7675imp 444 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
7751adantr 480 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 1 ∈ (1...𝑀))
78 fmuldfeq.9 . . . . . . . . . . . . 13 (𝜑𝑈:(1...𝑀)⟶𝑌)
7978, 51ffvelrnd 6268 . . . . . . . . . . . 12 (𝜑 → (𝑈‘1) ∈ 𝑌)
8079ancli 572 . . . . . . . . . . . 12 (𝜑 → (𝜑 ∧ (𝑈‘1) ∈ 𝑌))
81 eleq1 2676 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈‘1) → (𝑓𝑌 ↔ (𝑈‘1) ∈ 𝑌))
8281anbi2d 736 . . . . . . . . . . . . . 14 (𝑓 = (𝑈‘1) → ((𝜑𝑓𝑌) ↔ (𝜑 ∧ (𝑈‘1) ∈ 𝑌)))
83 feq1 5939 . . . . . . . . . . . . . 14 (𝑓 = (𝑈‘1) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘1):𝑇⟶ℝ))
8482, 83imbi12d 333 . . . . . . . . . . . . 13 (𝑓 = (𝑈‘1) → (((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ)))
85 fmuldfeq.10 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)
8685a1i 11 . . . . . . . . . . . . 13 (𝑓𝑌 → ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ))
8784, 86vtoclga 3245 . . . . . . . . . . . 12 ((𝑈‘1) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ))
8879, 80, 87sylc 63 . . . . . . . . . . 11 (𝜑 → (𝑈‘1):𝑇⟶ℝ)
8988ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝑈‘1)‘𝑡) ∈ ℝ)
90 fveq2 6103 . . . . . . . . . . . 12 (𝑖 = 1 → (𝑈𝑖) = (𝑈‘1))
9190fveq1d 6105 . . . . . . . . . . 11 (𝑖 = 1 → ((𝑈𝑖)‘𝑡) = ((𝑈‘1)‘𝑡))
92 eqid 2610 . . . . . . . . . . 11 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
9391, 92fvmptg 6189 . . . . . . . . . 10 ((1 ∈ (1...𝑀) ∧ ((𝑈‘1)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1) = ((𝑈‘1)‘𝑡))
9477, 89, 93syl2anc 691 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1) = ((𝑈‘1)‘𝑡))
9576, 94eqtrd 2644 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((𝑈‘1)‘𝑡))
96 seq1 12676 . . . . . . . . . 10 (1 ∈ ℤ → (seq1(𝑃, 𝑈)‘1) = (𝑈‘1))
9743, 96ax-mp 5 . . . . . . . . 9 (seq1(𝑃, 𝑈)‘1) = (𝑈‘1)
9897fveq1i 6104 . . . . . . . 8 ((seq1(𝑃, 𝑈)‘1)‘𝑡) = ((𝑈‘1)‘𝑡)
9995, 98syl6eqr 2662 . . . . . . 7 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
10045, 99syl5req 2657 . . . . . 6 ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))
1011003adant3 1074 . . . . 5 ((𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))
102 simp31 1090 . . . . . . 7 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝜑)
103 simp1 1054 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ ℕ)
104 simp33 1092 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → (𝑛 + 1) ∈ (1...𝑀))
105103, 104jca 553 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑀)))
106 elnnuz 11600 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
107106biimpi 205 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
108107anim1i 590 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑀)) → (𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)))
109 peano2fzr 12225 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)) → 𝑛 ∈ (1...𝑀))
110105, 108, 1093syl 18 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ (1...𝑀))
111 simp32 1091 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑡𝑇)
112 simp2 1055 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
113102, 111, 110, 112mp3and 1419 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
114110, 104, 1133jca 1235 . . . . . . 7 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
115 nfv 1830 . . . . . . . . 9 𝑓𝜑
116 nfv 1830 . . . . . . . . . 10 𝑓 𝑛 ∈ (1...𝑀)
117 nfv 1830 . . . . . . . . . 10 𝑓(𝑛 + 1) ∈ (1...𝑀)
118 nfcv 2751 . . . . . . . . . . . . . 14 𝑓1
119 fmuldfeq.3 . . . . . . . . . . . . . . 15 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
120 nfmpt21 6620 . . . . . . . . . . . . . . 15 𝑓(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
121119, 120nfcxfr 2749 . . . . . . . . . . . . . 14 𝑓𝑃
122 nfcv 2751 . . . . . . . . . . . . . 14 𝑓𝑈
123118, 121, 122nfseq 12673 . . . . . . . . . . . . 13 𝑓seq1(𝑃, 𝑈)
124 nfcv 2751 . . . . . . . . . . . . 13 𝑓𝑛
125123, 124nffv 6110 . . . . . . . . . . . 12 𝑓(seq1(𝑃, 𝑈)‘𝑛)
126 nfcv 2751 . . . . . . . . . . . 12 𝑓𝑡
127125, 126nffv 6110 . . . . . . . . . . 11 𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
128 nfcv 2751 . . . . . . . . . . 11 𝑓(seq1( · , (𝐹𝑡))‘𝑛)
129127, 128nfeq 2762 . . . . . . . . . 10 𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)
130116, 117, 129nf3an 1819 . . . . . . . . 9 𝑓(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
131115, 130nfan 1816 . . . . . . . 8 𝑓(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
132 nfv 1830 . . . . . . . . 9 𝑔𝜑
133 nfv 1830 . . . . . . . . . 10 𝑔 𝑛 ∈ (1...𝑀)
134 nfv 1830 . . . . . . . . . 10 𝑔(𝑛 + 1) ∈ (1...𝑀)
135 nfcv 2751 . . . . . . . . . . . . . 14 𝑔1
136 nfmpt22 6621 . . . . . . . . . . . . . . 15 𝑔(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
137119, 136nfcxfr 2749 . . . . . . . . . . . . . 14 𝑔𝑃
138 nfcv 2751 . . . . . . . . . . . . . 14 𝑔𝑈
139135, 137, 138nfseq 12673 . . . . . . . . . . . . 13 𝑔seq1(𝑃, 𝑈)
140 nfcv 2751 . . . . . . . . . . . . 13 𝑔𝑛
141139, 140nffv 6110 . . . . . . . . . . . 12 𝑔(seq1(𝑃, 𝑈)‘𝑛)
142 nfcv 2751 . . . . . . . . . . . 12 𝑔𝑡
143141, 142nffv 6110 . . . . . . . . . . 11 𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
144 nfcv 2751 . . . . . . . . . . 11 𝑔(seq1( · , (𝐹𝑡))‘𝑛)
145143, 144nfeq 2762 . . . . . . . . . 10 𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)
146133, 134, 145nf3an 1819 . . . . . . . . 9 𝑔(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
147132, 146nfan 1816 . . . . . . . 8 𝑔(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
148 fmuldfeq.2 . . . . . . . 8 𝑡𝑌
149 fmuldfeq.7 . . . . . . . . 9 (𝜑𝑇 ∈ V)
150149adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑇 ∈ V)
15178adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑈:(1...𝑀)⟶𝑌)
152 fmuldfeq.11 . . . . . . . . 9 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
1531523adant1r 1311 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) ∧ 𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
154 simpr1 1060 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑛 ∈ (1...𝑀))
155 simpr2 1061 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → (𝑛 + 1) ∈ (1...𝑀))
156 simpr3 1062 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
15785adantlr 747 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) ∧ 𝑓𝑌) → 𝑓:𝑇⟶ℝ)
158131, 147, 148, 119, 53, 150, 151, 153, 154, 155, 156, 157fmuldfeqlem1 38649 . . . . . . 7 (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) ∧ 𝑡𝑇) → ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1)))
159102, 114, 111, 158syl21anc 1317 . . . . . 6 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1)))
1601593exp 1256 . . . . 5 (𝑛 ∈ ℕ → (((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) → ((𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1)))))
16121, 28, 35, 42, 101, 160nnind 10915 . . . 4 (𝑀 ∈ ℕ → ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀)))
16214, 161mpcom 37 . . 3 ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
16313, 162mpd3an3 1417 . 2 ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
164 fmuldfeq.4 . . . 4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
165164fveq1i 6104 . . 3 (𝑋𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡)
166165a1i 11 . 2 ((𝜑𝑡𝑇) → (𝑋𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
167 simpr 476 . . 3 ((𝜑𝑡𝑇) → 𝑡𝑇)
168 elnnuz 11600 . . . . . 6 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ‘1))
1691, 168sylib 207 . . . . 5 (𝜑𝑀 ∈ (ℤ‘1))
170169adantr 480 . . . 4 ((𝜑𝑡𝑇) → 𝑀 ∈ (ℤ‘1))
171 fmuldfeq.1 . . . . . . . 8 𝑖𝜑
172171, 52nfan 1816 . . . . . . 7 𝑖(𝜑𝑡𝑇)
173 nfv 1830 . . . . . . 7 𝑖 𝑘 ∈ (1...𝑀)
174172, 173nfan 1816 . . . . . 6 𝑖((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀))
175 nfcv 2751 . . . . . . . 8 𝑖𝑘
17659, 175nffv 6110 . . . . . . 7 𝑖((𝐹𝑡)‘𝑘)
177176nfel1 2765 . . . . . 6 𝑖((𝐹𝑡)‘𝑘) ∈ ℝ
178174, 177nfim 1813 . . . . 5 𝑖(((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
179 eleq1 2676 . . . . . . 7 (𝑖 = 𝑘 → (𝑖 ∈ (1...𝑀) ↔ 𝑘 ∈ (1...𝑀)))
180179anbi2d 736 . . . . . 6 (𝑖 = 𝑘 → (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀))))
181 fveq2 6103 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘𝑘))
182181eleq1d 2672 . . . . . 6 (𝑖 = 𝑘 → (((𝐹𝑡)‘𝑖) ∈ ℝ ↔ ((𝐹𝑡)‘𝑘) ∈ ℝ))
183180, 182imbi12d 333 . . . . 5 (𝑖 = 𝑘 → ((((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ∈ ℝ) ↔ (((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)))
18473ad2antlr 759 . . . . . 6 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
185 simpr 476 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
18678ffvelrnda 6267 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
187 simpl 472 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
188187, 186jca 553 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈𝑖) ∈ 𝑌))
189 eleq1 2676 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → (𝑓𝑌 ↔ (𝑈𝑖) ∈ 𝑌))
190189anbi2d 736 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝑌) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝑌)))
191 feq1 5939 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
192190, 191imbi12d 333 . . . . . . . . . . . 12 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝑌) → (𝑈𝑖):𝑇⟶ℝ)))
193192, 86vtoclga 3245 . . . . . . . . . . 11 ((𝑈𝑖) ∈ 𝑌 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝑌) → (𝑈𝑖):𝑇⟶ℝ))
194186, 188, 193sylc 63 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
195194adantlr 747 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
196 simplr 788 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
197195, 196ffvelrnd 6268 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
19892fvmpt2 6200 . . . . . . . 8 ((𝑖 ∈ (1...𝑀) ∧ ((𝑈𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
199185, 197, 198syl2anc 691 . . . . . . 7 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
200199, 197eqeltrd 2688 . . . . . 6 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) ∈ ℝ)
201184, 200eqeltrd 2688 . . . . 5 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ∈ ℝ)
202178, 183, 201chvar 2250 . . . 4 (((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
203 remulcl 9900 . . . . 5 ((𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑘 · 𝑏) ∈ ℝ)
204203adantl 481 . . . 4 (((𝜑𝑡𝑇) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑘 · 𝑏) ∈ ℝ)
205170, 202, 204seqcl 12683 . . 3 ((𝜑𝑡𝑇) → (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ)
206 fmuldfeq.6 . . . 4 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
207206fvmpt2 6200 . . 3 ((𝑡𝑇 ∧ (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
208167, 205, 207syl2anc 691 . 2 ((𝜑𝑡𝑇) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
209163, 166, 2083eqtr4d 2654 1 ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wnf 1699  wcel 1977  wnfc 2738  Vcvv 3173   class class class wbr 4583  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  cr 9814  1c1 9816   + caddc 9818   · cmul 9820  cle 9954  cn 10897  cz 11254  cuz 11563  ...cfz 12197  seqcseq 12663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664
This theorem is referenced by:  stoweidlem42  38935  stoweidlem48  38941
  Copyright terms: Public domain W3C validator