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

Theorem fmuldfeq 42592
 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 11723 . . . . 5 (𝜑 → 1 ≤ 𝑀)
32adantr 485 . . . 4 ((𝜑𝑡𝑇) → 1 ≤ 𝑀)
4 nnre 11682 . . . . . 6 (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
5 leid 10775 . . . . . 6 (𝑀 ∈ ℝ → 𝑀𝑀)
61, 4, 53syl 18 . . . . 5 (𝜑𝑀𝑀)
76adantr 485 . . . 4 ((𝜑𝑡𝑇) → 𝑀𝑀)
81nnzd 12126 . . . . . 6 (𝜑𝑀 ∈ ℤ)
98adantr 485 . . . . 5 ((𝜑𝑡𝑇) → 𝑀 ∈ ℤ)
10 1zzd 12053 . . . . 5 ((𝜑𝑡𝑇) → 1 ∈ ℤ)
11 elfz 12946 . . . . 5 ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (1...𝑀) ↔ (1 ≤ 𝑀𝑀𝑀)))
129, 10, 9, 11syl3anc 1369 . . . 4 ((𝜑𝑡𝑇) → (𝑀 ∈ (1...𝑀) ↔ (1 ≤ 𝑀𝑀𝑀)))
133, 7, 12mpbir2and 713 . . 3 ((𝜑𝑡𝑇) → 𝑀 ∈ (1...𝑀))
1413ad2ant1 1131 . . . 4 ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → 𝑀 ∈ ℕ)
15 eleq1 2840 . . . . . . 7 (𝑚 = 1 → (𝑚 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
16153anbi3d 1440 . . . . . 6 (𝑚 = 1 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀))))
17 fveq2 6659 . . . . . . . 8 (𝑚 = 1 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘1))
1817fveq1d 6661 . . . . . . 7 (𝑚 = 1 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
19 fveq2 6659 . . . . . . 7 (𝑚 = 1 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘1))
2018, 19eqeq12d 2775 . . . . . 6 (𝑚 = 1 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1)))
2116, 20imbi12d 349 . . . . 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 2840 . . . . . . 7 (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑀) ↔ 𝑛 ∈ (1...𝑀)))
23223anbi3d 1440 . . . . . 6 (𝑚 = 𝑛 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇𝑛 ∈ (1...𝑀))))
24 fveq2 6659 . . . . . . . 8 (𝑚 = 𝑛 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑛))
2524fveq1d 6661 . . . . . . 7 (𝑚 = 𝑛 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡))
26 fveq2 6659 . . . . . . 7 (𝑚 = 𝑛 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘𝑛))
2725, 26eqeq12d 2775 . . . . . 6 (𝑚 = 𝑛 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
2823, 27imbi12d 349 . . . . 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 2840 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑚 ∈ (1...𝑀) ↔ (𝑛 + 1) ∈ (1...𝑀)))
30293anbi3d 1440 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))))
31 fveq2 6659 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘(𝑛 + 1)))
3231fveq1d 6661 . . . . . . 7 (𝑚 = (𝑛 + 1) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡))
33 fveq2 6659 . . . . . . 7 (𝑚 = (𝑛 + 1) → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1)))
3432, 33eqeq12d 2775 . . . . . 6 (𝑚 = (𝑛 + 1) → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1))))
3530, 34imbi12d 349 . . . . 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 2840 . . . . . . 7 (𝑚 = 𝑀 → (𝑚 ∈ (1...𝑀) ↔ 𝑀 ∈ (1...𝑀)))
37363anbi3d 1440 . . . . . 6 (𝑚 = 𝑀 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇𝑀 ∈ (1...𝑀))))
38 fveq2 6659 . . . . . . . 8 (𝑚 = 𝑀 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑀))
3938fveq1d 6661 . . . . . . 7 (𝑚 = 𝑀 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
40 fveq2 6659 . . . . . . 7 (𝑚 = 𝑀 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘𝑀))
4139, 40eqeq12d 2775 . . . . . 6 (𝑚 = 𝑀 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀)))
4237, 41imbi12d 349 . . . . 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 12052 . . . . . . . 8 1 ∈ ℤ
44 seq1 13432 . . . . . . . 8 (1 ∈ ℤ → (seq1( · , (𝐹𝑡))‘1) = ((𝐹𝑡)‘1))
4543, 44ax-mp 5 . . . . . . 7 (seq1( · , (𝐹𝑡))‘1) = ((𝐹𝑡)‘1)
46 1le1 11307 . . . . . . . . . . . . 13 1 ≤ 1
4746a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ≤ 1)
48 1zzd 12053 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ ℤ)
49 elfz 12946 . . . . . . . . . . . . 13 ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 ∈ (1...𝑀) ↔ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
5048, 48, 8, 49syl3anc 1369 . . . . . . . . . . . 12 (𝜑 → (1 ∈ (1...𝑀) ↔ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
5147, 2, 50mpbir2and 713 . . . . . . . . . . 11 (𝜑 → 1 ∈ (1...𝑀))
52 nfv 1916 . . . . . . . . . . . . 13 𝑖 𝑡𝑇
53 fmuldfeq.5 . . . . . . . . . . . . . . . . 17 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
54 nfcv 2920 . . . . . . . . . . . . . . . . . 18 𝑖𝑇
55 nfmpt1 5131 . . . . . . . . . . . . . . . . . 18 𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
5654, 55nfmpt 5130 . . . . . . . . . . . . . . . . 17 𝑖(𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
5753, 56nfcxfr 2918 . . . . . . . . . . . . . . . 16 𝑖𝐹
58 nfcv 2920 . . . . . . . . . . . . . . . 16 𝑖𝑡
5957, 58nffv 6669 . . . . . . . . . . . . . . 15 𝑖(𝐹𝑡)
60 nfcv 2920 . . . . . . . . . . . . . . 15 𝑖1
6159, 60nffv 6669 . . . . . . . . . . . . . 14 𝑖((𝐹𝑡)‘1)
62 nffvmpt1 6670 . . . . . . . . . . . . . 14 𝑖((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)
6361, 62nfeq 2933 . . . . . . . . . . . . 13 𝑖((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)
6452, 63nfim 1898 . . . . . . . . . . . 12 𝑖(𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
65 fveq2 6659 . . . . . . . . . . . . . 14 (𝑖 = 1 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘1))
66 fveq2 6659 . . . . . . . . . . . . . 14 (𝑖 = 1 → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
6765, 66eqeq12d 2775 . . . . . . . . . . . . 13 (𝑖 = 1 → (((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) ↔ ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)))
6867imbi2d 345 . . . . . . . . . . . 12 (𝑖 = 1 → ((𝑡𝑇 → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖)) ↔ (𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))))
69 ovex 7184 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ V
7069mptex 6978 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V
7153fvmpt2 6771 . . . . . . . . . . . . . 14 ((𝑡𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V) → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7270, 71mpan2 691 . . . . . . . . . . . . 13 (𝑡𝑇 → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7372fveq1d 6661 . . . . . . . . . . . 12 (𝑡𝑇 → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
7464, 68, 73vtoclg1f 3485 . . . . . . . . . . 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 411 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
77 eqid 2759 . . . . . . . . . 10 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
78 fveq2 6659 . . . . . . . . . . 11 (𝑖 = 1 → (𝑈𝑖) = (𝑈‘1))
7978fveq1d 6661 . . . . . . . . . 10 (𝑖 = 1 → ((𝑈𝑖)‘𝑡) = ((𝑈‘1)‘𝑡))
8051adantr 485 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 1 ∈ (1...𝑀))
81 fmuldfeq.9 . . . . . . . . . . . . 13 (𝜑𝑈:(1...𝑀)⟶𝑌)
8281, 51ffvelrnd 6844 . . . . . . . . . . . 12 (𝜑 → (𝑈‘1) ∈ 𝑌)
8382ancli 553 . . . . . . . . . . . 12 (𝜑 → (𝜑 ∧ (𝑈‘1) ∈ 𝑌))
84 eleq1 2840 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈‘1) → (𝑓𝑌 ↔ (𝑈‘1) ∈ 𝑌))
8584anbi2d 632 . . . . . . . . . . . . . 14 (𝑓 = (𝑈‘1) → ((𝜑𝑓𝑌) ↔ (𝜑 ∧ (𝑈‘1) ∈ 𝑌)))
86 feq1 6480 . . . . . . . . . . . . . 14 (𝑓 = (𝑈‘1) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘1):𝑇⟶ℝ))
8785, 86imbi12d 349 . . . . . . . . . . . . 13 (𝑓 = (𝑈‘1) → (((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ)))
88 fmuldfeq.10 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)
8988a1i 11 . . . . . . . . . . . . 13 (𝑓𝑌 → ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ))
9087, 89vtoclga 3493 . . . . . . . . . . . 12 ((𝑈‘1) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ))
9182, 83, 90sylc 65 . . . . . . . . . . 11 (𝜑 → (𝑈‘1):𝑇⟶ℝ)
9291ffvelrnda 6843 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝑈‘1)‘𝑡) ∈ ℝ)
9377, 79, 80, 92fvmptd3 6783 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1) = ((𝑈‘1)‘𝑡))
9476, 93eqtrd 2794 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((𝑈‘1)‘𝑡))
95 seq1 13432 . . . . . . . . . 10 (1 ∈ ℤ → (seq1(𝑃, 𝑈)‘1) = (𝑈‘1))
9643, 95ax-mp 5 . . . . . . . . 9 (seq1(𝑃, 𝑈)‘1) = (𝑈‘1)
9796fveq1i 6660 . . . . . . . 8 ((seq1(𝑃, 𝑈)‘1)‘𝑡) = ((𝑈‘1)‘𝑡)
9894, 97eqtr4di 2812 . . . . . . 7 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
9945, 98syl5req 2807 . . . . . 6 ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))
100993adant3 1130 . . . . 5 ((𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))
101 simp31 1207 . . . . . . 7 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝜑)
102 simp1 1134 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ ℕ)
103 simp33 1209 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → (𝑛 + 1) ∈ (1...𝑀))
104102, 103jca 516 . . . . . . . . 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)))
105 elnnuz 12323 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
106105biimpi 219 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
107106anim1i 618 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑀)) → (𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)))
108 peano2fzr 12970 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)) → 𝑛 ∈ (1...𝑀))
109104, 107, 1083syl 18 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ (1...𝑀))
110 simp32 1208 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑡𝑇)
111 simp2 1135 . . . . . . . . 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)))
112101, 110, 109, 111mp3and 1462 . . . . . . . 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))
113109, 103, 1123jca 1126 . . . . . . 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)))
114 nfv 1916 . . . . . . . . 9 𝑓𝜑
115 nfv 1916 . . . . . . . . . 10 𝑓 𝑛 ∈ (1...𝑀)
116 nfv 1916 . . . . . . . . . 10 𝑓(𝑛 + 1) ∈ (1...𝑀)
117 nfcv 2920 . . . . . . . . . . . . . 14 𝑓1
118 fmuldfeq.3 . . . . . . . . . . . . . . 15 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
119 nfmpo1 7229 . . . . . . . . . . . . . . 15 𝑓(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
120118, 119nfcxfr 2918 . . . . . . . . . . . . . 14 𝑓𝑃
121 nfcv 2920 . . . . . . . . . . . . . 14 𝑓𝑈
122117, 120, 121nfseq 13429 . . . . . . . . . . . . 13 𝑓seq1(𝑃, 𝑈)
123 nfcv 2920 . . . . . . . . . . . . 13 𝑓𝑛
124122, 123nffv 6669 . . . . . . . . . . . 12 𝑓(seq1(𝑃, 𝑈)‘𝑛)
125 nfcv 2920 . . . . . . . . . . . 12 𝑓𝑡
126124, 125nffv 6669 . . . . . . . . . . 11 𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
127 nfcv 2920 . . . . . . . . . . 11 𝑓(seq1( · , (𝐹𝑡))‘𝑛)
128126, 127nfeq 2933 . . . . . . . . . 10 𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)
129115, 116, 128nf3an 1903 . . . . . . . . 9 𝑓(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
130114, 129nfan 1901 . . . . . . . 8 𝑓(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
131 nfv 1916 . . . . . . . . 9 𝑔𝜑
132 nfv 1916 . . . . . . . . . 10 𝑔 𝑛 ∈ (1...𝑀)
133 nfv 1916 . . . . . . . . . 10 𝑔(𝑛 + 1) ∈ (1...𝑀)
134 nfcv 2920 . . . . . . . . . . . . . 14 𝑔1
135 nfmpo2 7230 . . . . . . . . . . . . . . 15 𝑔(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
136118, 135nfcxfr 2918 . . . . . . . . . . . . . 14 𝑔𝑃
137 nfcv 2920 . . . . . . . . . . . . . 14 𝑔𝑈
138134, 136, 137nfseq 13429 . . . . . . . . . . . . 13 𝑔seq1(𝑃, 𝑈)
139 nfcv 2920 . . . . . . . . . . . . 13 𝑔𝑛
140138, 139nffv 6669 . . . . . . . . . . . 12 𝑔(seq1(𝑃, 𝑈)‘𝑛)
141 nfcv 2920 . . . . . . . . . . . 12 𝑔𝑡
142140, 141nffv 6669 . . . . . . . . . . 11 𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
143 nfcv 2920 . . . . . . . . . . 11 𝑔(seq1( · , (𝐹𝑡))‘𝑛)
144142, 143nfeq 2933 . . . . . . . . . 10 𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)
145132, 133, 144nf3an 1903 . . . . . . . . 9 𝑔(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
146131, 145nfan 1901 . . . . . . . 8 𝑔(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
147 fmuldfeq.2 . . . . . . . 8 𝑡𝑌
148 fmuldfeq.7 . . . . . . . . 9 (𝜑𝑇 ∈ V)
149148adantr 485 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑇 ∈ V)
15081adantr 485 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑈:(1...𝑀)⟶𝑌)
151 fmuldfeq.11 . . . . . . . . 9 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
1521513adant1r 1175 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) ∧ 𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
153 simpr1 1192 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑛 ∈ (1...𝑀))
154 simpr2 1193 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → (𝑛 + 1) ∈ (1...𝑀))
155 simpr3 1194 . . . . . . . 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))
15688adantlr 715 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) ∧ 𝑓𝑌) → 𝑓:𝑇⟶ℝ)
157130, 146, 147, 118, 53, 149, 150, 152, 153, 154, 155, 156fmuldfeqlem1 42591 . . . . . . 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)))
158101, 113, 110, 157syl21anc 837 . . . . . 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)))
1591583exp 1117 . . . . 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)))))
16021, 28, 35, 42, 100, 159nnind 11693 . . . 4 (𝑀 ∈ ℕ → ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀)))
16114, 160mpcom 38 . . 3 ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
16213, 161mpd3an3 1460 . 2 ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
163 fmuldfeq.4 . . . 4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
164163fveq1i 6660 . . 3 (𝑋𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡)
165164a1i 11 . 2 ((𝜑𝑡𝑇) → (𝑋𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
166 simpr 489 . . 3 ((𝜑𝑡𝑇) → 𝑡𝑇)
167 elnnuz 12323 . . . . . 6 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ‘1))
1681, 167sylib 221 . . . . 5 (𝜑𝑀 ∈ (ℤ‘1))
169168adantr 485 . . . 4 ((𝜑𝑡𝑇) → 𝑀 ∈ (ℤ‘1))
170 fmuldfeq.1 . . . . . . . 8 𝑖𝜑
171170, 52nfan 1901 . . . . . . 7 𝑖(𝜑𝑡𝑇)
172 nfv 1916 . . . . . . 7 𝑖 𝑘 ∈ (1...𝑀)
173171, 172nfan 1901 . . . . . 6 𝑖((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀))
174 nfcv 2920 . . . . . . . 8 𝑖𝑘
17559, 174nffv 6669 . . . . . . 7 𝑖((𝐹𝑡)‘𝑘)
176175nfel1 2936 . . . . . 6 𝑖((𝐹𝑡)‘𝑘) ∈ ℝ
177173, 176nfim 1898 . . . . 5 𝑖(((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
178 eleq1 2840 . . . . . . 7 (𝑖 = 𝑘 → (𝑖 ∈ (1...𝑀) ↔ 𝑘 ∈ (1...𝑀)))
179178anbi2d 632 . . . . . 6 (𝑖 = 𝑘 → (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀))))
180 fveq2 6659 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘𝑘))
181180eleq1d 2837 . . . . . 6 (𝑖 = 𝑘 → (((𝐹𝑡)‘𝑖) ∈ ℝ ↔ ((𝐹𝑡)‘𝑘) ∈ ℝ))
182179, 181imbi12d 349 . . . . 5 (𝑖 = 𝑘 → ((((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ∈ ℝ) ↔ (((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)))
18373ad2antlr 727 . . . . . 6 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
184 simpr 489 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
18581ffvelrnda 6843 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
186 simpl 487 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
187186, 185jca 516 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈𝑖) ∈ 𝑌))
188 eleq1 2840 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → (𝑓𝑌 ↔ (𝑈𝑖) ∈ 𝑌))
189188anbi2d 632 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝑌) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝑌)))
190 feq1 6480 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
191189, 190imbi12d 349 . . . . . . . . . . . 12 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝑌) → (𝑈𝑖):𝑇⟶ℝ)))
192191, 89vtoclga 3493 . . . . . . . . . . 11 ((𝑈𝑖) ∈ 𝑌 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝑌) → (𝑈𝑖):𝑇⟶ℝ))
193185, 187, 192sylc 65 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
194193adantlr 715 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
195 simplr 769 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
196194, 195ffvelrnd 6844 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
19777fvmpt2 6771 . . . . . . . 8 ((𝑖 ∈ (1...𝑀) ∧ ((𝑈𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
198184, 196, 197syl2anc 588 . . . . . . 7 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
199198, 196eqeltrd 2853 . . . . . 6 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) ∈ ℝ)
200183, 199eqeltrd 2853 . . . . 5 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ∈ ℝ)
201177, 182, 200chvarfv 2241 . . . 4 (((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
202 remulcl 10661 . . . . 5 ((𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑘 · 𝑏) ∈ ℝ)
203202adantl 486 . . . 4 (((𝜑𝑡𝑇) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑘 · 𝑏) ∈ ℝ)
204169, 201, 203seqcl 13441 . . 3 ((𝜑𝑡𝑇) → (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ)
205 fmuldfeq.6 . . . 4 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
206205fvmpt2 6771 . . 3 ((𝑡𝑇 ∧ (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
207166, 204, 206syl2anc 588 . 2 ((𝜑𝑡𝑇) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
208162, 165, 2073eqtr4d 2804 1 ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2112  Ⅎwnfc 2900  Vcvv 3410   class class class wbr 5033   ↦ cmpt 5113  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153  ℝcr 10575  1c1 10577   + caddc 10579   · cmul 10581   ≤ cle 10715  ℕcn 11675  ℤcz 12021  ℤ≥cuz 12283  ...cfz 12940  seqcseq 13419 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10632  ax-resscn 10633  ax-1cn 10634  ax-icn 10635  ax-addcl 10636  ax-addrcl 10637  ax-mulcl 10638  ax-mulrcl 10639  ax-mulcom 10640  ax-addass 10641  ax-mulass 10642  ax-distr 10643  ax-i2m1 10644  ax-1ne0 10645  ax-1rid 10646  ax-rnegex 10647  ax-rrecex 10648  ax-cnre 10649  ax-pre-lttri 10650  ax-pre-lttrn 10651  ax-pre-ltadd 10652  ax-pre-mulgt0 10653 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10716  df-mnf 10717  df-xr 10718  df-ltxr 10719  df-le 10720  df-sub 10911  df-neg 10912  df-nn 11676  df-n0 11936  df-z 12022  df-uz 12284  df-fz 12941  df-seq 13420 This theorem is referenced by:  stoweidlem42  43051  stoweidlem48  43057
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