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Theorem fmuldfeq 40333
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 11265 . . . . 5 (𝜑 → 1 ≤ 𝑀)
32adantr 466 . . . 4 ((𝜑𝑡𝑇) → 1 ≤ 𝑀)
4 nnre 11229 . . . . . 6 (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
5 leid 10335 . . . . . 6 (𝑀 ∈ ℝ → 𝑀𝑀)
61, 4, 53syl 18 . . . . 5 (𝜑𝑀𝑀)
76adantr 466 . . . 4 ((𝜑𝑡𝑇) → 𝑀𝑀)
81nnzd 11683 . . . . . 6 (𝜑𝑀 ∈ ℤ)
98adantr 466 . . . . 5 ((𝜑𝑡𝑇) → 𝑀 ∈ ℤ)
10 1zzd 11610 . . . . 5 ((𝜑𝑡𝑇) → 1 ∈ ℤ)
11 elfz 12539 . . . . 5 ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 ∈ (1...𝑀) ↔ (1 ≤ 𝑀𝑀𝑀)))
129, 10, 9, 11syl3anc 1476 . . . 4 ((𝜑𝑡𝑇) → (𝑀 ∈ (1...𝑀) ↔ (1 ≤ 𝑀𝑀𝑀)))
133, 7, 12mpbir2and 684 . . 3 ((𝜑𝑡𝑇) → 𝑀 ∈ (1...𝑀))
1413ad2ant1 1127 . . . 4 ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → 𝑀 ∈ ℕ)
15 eleq1 2838 . . . . . . 7 (𝑚 = 1 → (𝑚 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
16153anbi3d 1553 . . . . . 6 (𝑚 = 1 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀))))
17 fveq2 6332 . . . . . . . 8 (𝑚 = 1 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘1))
1817fveq1d 6334 . . . . . . 7 (𝑚 = 1 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
19 fveq2 6332 . . . . . . 7 (𝑚 = 1 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘1))
2018, 19eqeq12d 2786 . . . . . 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 2838 . . . . . . 7 (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑀) ↔ 𝑛 ∈ (1...𝑀)))
23223anbi3d 1553 . . . . . 6 (𝑚 = 𝑛 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇𝑛 ∈ (1...𝑀))))
24 fveq2 6332 . . . . . . . 8 (𝑚 = 𝑛 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑛))
2524fveq1d 6334 . . . . . . 7 (𝑚 = 𝑛 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡))
26 fveq2 6332 . . . . . . 7 (𝑚 = 𝑛 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘𝑛))
2725, 26eqeq12d 2786 . . . . . 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 2838 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑚 ∈ (1...𝑀) ↔ (𝑛 + 1) ∈ (1...𝑀)))
30293anbi3d 1553 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))))
31 fveq2 6332 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘(𝑛 + 1)))
3231fveq1d 6334 . . . . . . 7 (𝑚 = (𝑛 + 1) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡))
33 fveq2 6332 . . . . . . 7 (𝑚 = (𝑛 + 1) → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1)))
3432, 33eqeq12d 2786 . . . . . 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 2838 . . . . . . 7 (𝑚 = 𝑀 → (𝑚 ∈ (1...𝑀) ↔ 𝑀 ∈ (1...𝑀)))
37363anbi3d 1553 . . . . . 6 (𝑚 = 𝑀 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇𝑀 ∈ (1...𝑀))))
38 fveq2 6332 . . . . . . . 8 (𝑚 = 𝑀 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑀))
3938fveq1d 6334 . . . . . . 7 (𝑚 = 𝑀 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
40 fveq2 6332 . . . . . . 7 (𝑚 = 𝑀 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘𝑀))
4139, 40eqeq12d 2786 . . . . . 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 11609 . . . . . . . 8 1 ∈ ℤ
44 seq1 13021 . . . . . . . 8 (1 ∈ ℤ → (seq1( · , (𝐹𝑡))‘1) = ((𝐹𝑡)‘1))
4543, 44ax-mp 5 . . . . . . 7 (seq1( · , (𝐹𝑡))‘1) = ((𝐹𝑡)‘1)
46 1le1 10857 . . . . . . . . . . . . 13 1 ≤ 1
4746a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ≤ 1)
48 1zzd 11610 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ ℤ)
49 elfz 12539 . . . . . . . . . . . . 13 ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 ∈ (1...𝑀) ↔ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
5048, 48, 8, 49syl3anc 1476 . . . . . . . . . . . 12 (𝜑 → (1 ∈ (1...𝑀) ↔ (1 ≤ 1 ∧ 1 ≤ 𝑀)))
5147, 2, 50mpbir2and 684 . . . . . . . . . . 11 (𝜑 → 1 ∈ (1...𝑀))
52 nfv 1995 . . . . . . . . . . . . 13 𝑖 𝑡𝑇
53 fmuldfeq.5 . . . . . . . . . . . . . . . . 17 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
54 nfcv 2913 . . . . . . . . . . . . . . . . . 18 𝑖𝑇
55 nfmpt1 4881 . . . . . . . . . . . . . . . . . 18 𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
5654, 55nfmpt 4880 . . . . . . . . . . . . . . . . 17 𝑖(𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
5753, 56nfcxfr 2911 . . . . . . . . . . . . . . . 16 𝑖𝐹
58 nfcv 2913 . . . . . . . . . . . . . . . 16 𝑖𝑡
5957, 58nffv 6339 . . . . . . . . . . . . . . 15 𝑖(𝐹𝑡)
60 nfcv 2913 . . . . . . . . . . . . . . 15 𝑖1
6159, 60nffv 6339 . . . . . . . . . . . . . 14 𝑖((𝐹𝑡)‘1)
62 nffvmpt1 6340 . . . . . . . . . . . . . 14 𝑖((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)
6361, 62nfeq 2925 . . . . . . . . . . . . 13 𝑖((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)
6452, 63nfim 1977 . . . . . . . . . . . 12 𝑖(𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
65 fveq2 6332 . . . . . . . . . . . . . 14 (𝑖 = 1 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘1))
66 fveq2 6332 . . . . . . . . . . . . . 14 (𝑖 = 1 → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
6765, 66eqeq12d 2786 . . . . . . . . . . . . 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 6823 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ V
7069mptex 6630 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V
7153fvmpt2 6433 . . . . . . . . . . . . . 14 ((𝑡𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V) → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7270, 71mpan2 663 . . . . . . . . . . . . 13 (𝑡𝑇 → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7372fveq1d 6334 . . . . . . . . . . . 12 (𝑡𝑇 → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
7464, 68, 73vtoclg1f 3416 . . . . . . . . . . 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 393 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
7751adantr 466 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 1 ∈ (1...𝑀))
78 fmuldfeq.9 . . . . . . . . . . . . 13 (𝜑𝑈:(1...𝑀)⟶𝑌)
7978, 51ffvelrnd 6503 . . . . . . . . . . . 12 (𝜑 → (𝑈‘1) ∈ 𝑌)
8079ancli 530 . . . . . . . . . . . 12 (𝜑 → (𝜑 ∧ (𝑈‘1) ∈ 𝑌))
81 eleq1 2838 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈‘1) → (𝑓𝑌 ↔ (𝑈‘1) ∈ 𝑌))
8281anbi2d 606 . . . . . . . . . . . . . 14 (𝑓 = (𝑈‘1) → ((𝜑𝑓𝑌) ↔ (𝜑 ∧ (𝑈‘1) ∈ 𝑌)))
83 feq1 6166 . . . . . . . . . . . . . 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 3423 . . . . . . . . . . . 12 ((𝑈‘1) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ))
8879, 80, 87sylc 65 . . . . . . . . . . 11 (𝜑 → (𝑈‘1):𝑇⟶ℝ)
8988ffvelrnda 6502 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝑈‘1)‘𝑡) ∈ ℝ)
90 fveq2 6332 . . . . . . . . . . . 12 (𝑖 = 1 → (𝑈𝑖) = (𝑈‘1))
9190fveq1d 6334 . . . . . . . . . . 11 (𝑖 = 1 → ((𝑈𝑖)‘𝑡) = ((𝑈‘1)‘𝑡))
92 eqid 2771 . . . . . . . . . . 11 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
9391, 92fvmptg 6422 . . . . . . . . . 10 ((1 ∈ (1...𝑀) ∧ ((𝑈‘1)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1) = ((𝑈‘1)‘𝑡))
9477, 89, 93syl2anc 565 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1) = ((𝑈‘1)‘𝑡))
9576, 94eqtrd 2805 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((𝑈‘1)‘𝑡))
96 seq1 13021 . . . . . . . . . 10 (1 ∈ ℤ → (seq1(𝑃, 𝑈)‘1) = (𝑈‘1))
9743, 96ax-mp 5 . . . . . . . . 9 (seq1(𝑃, 𝑈)‘1) = (𝑈‘1)
9897fveq1i 6333 . . . . . . . 8 ((seq1(𝑃, 𝑈)‘1)‘𝑡) = ((𝑈‘1)‘𝑡)
9995, 98syl6eqr 2823 . . . . . . 7 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
10045, 99syl5req 2818 . . . . . 6 ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))
1011003adant3 1126 . . . . 5 ((𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))
102 simp31 1251 . . . . . . 7 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝜑)
103 simp1 1130 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ ℕ)
104 simp33 1253 . . . . . . . . . 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 495 . . . . . . . . 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 11926 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
107106biimpi 206 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
108107anim1i 594 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑀)) → (𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)))
109 peano2fzr 12561 . . . . . . . . 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 1252 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑡𝑇)
112 simp2 1131 . . . . . . . . 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 1575 . . . . . . . 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 1122 . . . . . . 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 1995 . . . . . . . . 9 𝑓𝜑
116 nfv 1995 . . . . . . . . . 10 𝑓 𝑛 ∈ (1...𝑀)
117 nfv 1995 . . . . . . . . . 10 𝑓(𝑛 + 1) ∈ (1...𝑀)
118 nfcv 2913 . . . . . . . . . . . . . 14 𝑓1
119 fmuldfeq.3 . . . . . . . . . . . . . . 15 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
120 nfmpt21 6869 . . . . . . . . . . . . . . 15 𝑓(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
121119, 120nfcxfr 2911 . . . . . . . . . . . . . 14 𝑓𝑃
122 nfcv 2913 . . . . . . . . . . . . . 14 𝑓𝑈
123118, 121, 122nfseq 13018 . . . . . . . . . . . . 13 𝑓seq1(𝑃, 𝑈)
124 nfcv 2913 . . . . . . . . . . . . 13 𝑓𝑛
125123, 124nffv 6339 . . . . . . . . . . . 12 𝑓(seq1(𝑃, 𝑈)‘𝑛)
126 nfcv 2913 . . . . . . . . . . . 12 𝑓𝑡
127125, 126nffv 6339 . . . . . . . . . . 11 𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
128 nfcv 2913 . . . . . . . . . . 11 𝑓(seq1( · , (𝐹𝑡))‘𝑛)
129127, 128nfeq 2925 . . . . . . . . . 10 𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)
130116, 117, 129nf3an 1983 . . . . . . . . 9 𝑓(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
131115, 130nfan 1980 . . . . . . . 8 𝑓(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
132 nfv 1995 . . . . . . . . 9 𝑔𝜑
133 nfv 1995 . . . . . . . . . 10 𝑔 𝑛 ∈ (1...𝑀)
134 nfv 1995 . . . . . . . . . 10 𝑔(𝑛 + 1) ∈ (1...𝑀)
135 nfcv 2913 . . . . . . . . . . . . . 14 𝑔1
136 nfmpt22 6870 . . . . . . . . . . . . . . 15 𝑔(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
137119, 136nfcxfr 2911 . . . . . . . . . . . . . 14 𝑔𝑃
138 nfcv 2913 . . . . . . . . . . . . . 14 𝑔𝑈
139135, 137, 138nfseq 13018 . . . . . . . . . . . . 13 𝑔seq1(𝑃, 𝑈)
140 nfcv 2913 . . . . . . . . . . . . 13 𝑔𝑛
141139, 140nffv 6339 . . . . . . . . . . . 12 𝑔(seq1(𝑃, 𝑈)‘𝑛)
142 nfcv 2913 . . . . . . . . . . . 12 𝑔𝑡
143141, 142nffv 6339 . . . . . . . . . . 11 𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
144 nfcv 2913 . . . . . . . . . . 11 𝑔(seq1( · , (𝐹𝑡))‘𝑛)
145143, 144nfeq 2925 . . . . . . . . . 10 𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)
146133, 134, 145nf3an 1983 . . . . . . . . 9 𝑔(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
147132, 146nfan 1980 . . . . . . . 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 466 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑇 ∈ V)
15178adantr 466 . . . . . . . 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 1187 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) ∧ 𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
154 simpr1 1233 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑛 ∈ (1...𝑀))
155 simpr2 1235 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → (𝑛 + 1) ∈ (1...𝑀))
156 simpr3 1237 . . . . . . . 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 686 . . . . . . . 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 40332 . . . . . . 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 1475 . . . . . 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 1112 . . . . 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 11240 . . . 4 (𝑀 ∈ ℕ → ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀)))
16214, 161mpcom 38 . . 3 ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
16313, 162mpd3an3 1573 . 2 ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
164 fmuldfeq.4 . . . 4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
165164fveq1i 6333 . . 3 (𝑋𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡)
166165a1i 11 . 2 ((𝜑𝑡𝑇) → (𝑋𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
167 simpr 471 . . 3 ((𝜑𝑡𝑇) → 𝑡𝑇)
168 elnnuz 11926 . . . . . 6 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ‘1))
1691, 168sylib 208 . . . . 5 (𝜑𝑀 ∈ (ℤ‘1))
170169adantr 466 . . . 4 ((𝜑𝑡𝑇) → 𝑀 ∈ (ℤ‘1))
171 fmuldfeq.1 . . . . . . . 8 𝑖𝜑
172171, 52nfan 1980 . . . . . . 7 𝑖(𝜑𝑡𝑇)
173 nfv 1995 . . . . . . 7 𝑖 𝑘 ∈ (1...𝑀)
174172, 173nfan 1980 . . . . . 6 𝑖((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀))
175 nfcv 2913 . . . . . . . 8 𝑖𝑘
17659, 175nffv 6339 . . . . . . 7 𝑖((𝐹𝑡)‘𝑘)
177176nfel1 2928 . . . . . 6 𝑖((𝐹𝑡)‘𝑘) ∈ ℝ
178174, 177nfim 1977 . . . . 5 𝑖(((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
179 eleq1 2838 . . . . . . 7 (𝑖 = 𝑘 → (𝑖 ∈ (1...𝑀) ↔ 𝑘 ∈ (1...𝑀)))
180179anbi2d 606 . . . . . 6 (𝑖 = 𝑘 → (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀))))
181 fveq2 6332 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘𝑘))
182181eleq1d 2835 . . . . . 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 698 . . . . . 6 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
185 simpr 471 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
18678ffvelrnda 6502 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
187 simpl 468 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
188187, 186jca 495 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈𝑖) ∈ 𝑌))
189 eleq1 2838 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → (𝑓𝑌 ↔ (𝑈𝑖) ∈ 𝑌))
190189anbi2d 606 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝑌) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝑌)))
191 feq1 6166 . . . . . . . . . . . . 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 3423 . . . . . . . . . . 11 ((𝑈𝑖) ∈ 𝑌 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝑌) → (𝑈𝑖):𝑇⟶ℝ))
194186, 188, 193sylc 65 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
195194adantlr 686 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
196 simplr 744 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
197195, 196ffvelrnd 6503 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
19892fvmpt2 6433 . . . . . . . 8 ((𝑖 ∈ (1...𝑀) ∧ ((𝑈𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
199185, 197, 198syl2anc 565 . . . . . . 7 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
200199, 197eqeltrd 2850 . . . . . 6 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) ∈ ℝ)
201184, 200eqeltrd 2850 . . . . 5 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ∈ ℝ)
202178, 183, 201chvar 2424 . . . 4 (((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
203 remulcl 10223 . . . . 5 ((𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑘 · 𝑏) ∈ ℝ)
204203adantl 467 . . . 4 (((𝜑𝑡𝑇) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑘 · 𝑏) ∈ ℝ)
205170, 202, 204seqcl 13028 . . 3 ((𝜑𝑡𝑇) → (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ)
206 fmuldfeq.6 . . . 4 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
207206fvmpt2 6433 . . 3 ((𝑡𝑇 ∧ (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
208167, 205, 207syl2anc 565 . 2 ((𝜑𝑡𝑇) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
209163, 166, 2083eqtr4d 2815 1 ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wnf 1856  wcel 2145  wnfc 2900  Vcvv 3351   class class class wbr 4786  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  cr 10137  1c1 10139   + caddc 10141   · cmul 10143  cle 10277  cn 11222  cz 11579  cuz 11888  ...cfz 12533  seqcseq 13008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-seq 13009
This theorem is referenced by:  stoweidlem42  40776  stoweidlem48  40782
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