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Theorem fmuldfeq 45598
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 1zzd 12648 . . . 4 ((𝜑𝑡𝑇) → 1 ∈ ℤ)
2 fmuldfeq.8 . . . . . 6 (𝜑𝑀 ∈ ℕ)
32nnzd 12640 . . . . 5 (𝜑𝑀 ∈ ℤ)
43adantr 480 . . . 4 ((𝜑𝑡𝑇) → 𝑀 ∈ ℤ)
52nnge1d 12314 . . . . 5 (𝜑 → 1 ≤ 𝑀)
65adantr 480 . . . 4 ((𝜑𝑡𝑇) → 1 ≤ 𝑀)
7 nnre 12273 . . . . . 6 (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
8 leid 11357 . . . . . 6 (𝑀 ∈ ℝ → 𝑀𝑀)
92, 7, 83syl 18 . . . . 5 (𝜑𝑀𝑀)
109adantr 480 . . . 4 ((𝜑𝑡𝑇) → 𝑀𝑀)
111, 4, 4, 6, 10elfzd 13555 . . 3 ((𝜑𝑡𝑇) → 𝑀 ∈ (1...𝑀))
1223ad2ant1 1134 . . . 4 ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → 𝑀 ∈ ℕ)
13 eleq1 2829 . . . . . . 7 (𝑚 = 1 → (𝑚 ∈ (1...𝑀) ↔ 1 ∈ (1...𝑀)))
14133anbi3d 1444 . . . . . 6 (𝑚 = 1 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀))))
15 fveq2 6906 . . . . . . . 8 (𝑚 = 1 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘1))
1615fveq1d 6908 . . . . . . 7 (𝑚 = 1 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
17 fveq2 6906 . . . . . . 7 (𝑚 = 1 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘1))
1816, 17eqeq12d 2753 . . . . . 6 (𝑚 = 1 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1)))
1914, 18imbi12d 344 . . . . 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))))
20 eleq1 2829 . . . . . . 7 (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑀) ↔ 𝑛 ∈ (1...𝑀)))
21203anbi3d 1444 . . . . . 6 (𝑚 = 𝑛 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇𝑛 ∈ (1...𝑀))))
22 fveq2 6906 . . . . . . . 8 (𝑚 = 𝑛 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑛))
2322fveq1d 6908 . . . . . . 7 (𝑚 = 𝑛 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡))
24 fveq2 6906 . . . . . . 7 (𝑚 = 𝑛 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘𝑛))
2523, 24eqeq12d 2753 . . . . . 6 (𝑚 = 𝑛 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
2621, 25imbi12d 344 . . . . 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))))
27 eleq1 2829 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑚 ∈ (1...𝑀) ↔ (𝑛 + 1) ∈ (1...𝑀)))
28273anbi3d 1444 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))))
29 fveq2 6906 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘(𝑛 + 1)))
3029fveq1d 6908 . . . . . . 7 (𝑚 = (𝑛 + 1) → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡))
31 fveq2 6906 . . . . . . 7 (𝑚 = (𝑛 + 1) → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1)))
3230, 31eqeq12d 2753 . . . . . 6 (𝑚 = (𝑛 + 1) → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘(𝑛 + 1))‘𝑡) = (seq1( · , (𝐹𝑡))‘(𝑛 + 1))))
3328, 32imbi12d 344 . . . . 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)))))
34 eleq1 2829 . . . . . . 7 (𝑚 = 𝑀 → (𝑚 ∈ (1...𝑀) ↔ 𝑀 ∈ (1...𝑀)))
35343anbi3d 1444 . . . . . 6 (𝑚 = 𝑀 → ((𝜑𝑡𝑇𝑚 ∈ (1...𝑀)) ↔ (𝜑𝑡𝑇𝑀 ∈ (1...𝑀))))
36 fveq2 6906 . . . . . . . 8 (𝑚 = 𝑀 → (seq1(𝑃, 𝑈)‘𝑚) = (seq1(𝑃, 𝑈)‘𝑀))
3736fveq1d 6908 . . . . . . 7 (𝑚 = 𝑀 → ((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
38 fveq2 6906 . . . . . . 7 (𝑚 = 𝑀 → (seq1( · , (𝐹𝑡))‘𝑚) = (seq1( · , (𝐹𝑡))‘𝑀))
3937, 38eqeq12d 2753 . . . . . 6 (𝑚 = 𝑀 → (((seq1(𝑃, 𝑈)‘𝑚)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑚) ↔ ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀)))
4035, 39imbi12d 344 . . . . 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))))
41 1z 12647 . . . . . . . 8 1 ∈ ℤ
42 seq1 14055 . . . . . . . 8 (1 ∈ ℤ → (seq1( · , (𝐹𝑡))‘1) = ((𝐹𝑡)‘1))
4341, 42ax-mp 5 . . . . . . 7 (seq1( · , (𝐹𝑡))‘1) = ((𝐹𝑡)‘1)
44 1zzd 12648 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℤ)
45 1le1 11891 . . . . . . . . . . . . 13 1 ≤ 1
4645a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ≤ 1)
4744, 3, 44, 46, 5elfzd 13555 . . . . . . . . . . 11 (𝜑 → 1 ∈ (1...𝑀))
48 nfv 1914 . . . . . . . . . . . . 13 𝑖 𝑡𝑇
49 fmuldfeq.5 . . . . . . . . . . . . . . . . 17 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
50 nfcv 2905 . . . . . . . . . . . . . . . . . 18 𝑖𝑇
51 nfmpt1 5250 . . . . . . . . . . . . . . . . . 18 𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
5250, 51nfmpt 5249 . . . . . . . . . . . . . . . . 17 𝑖(𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
5349, 52nfcxfr 2903 . . . . . . . . . . . . . . . 16 𝑖𝐹
54 nfcv 2905 . . . . . . . . . . . . . . . 16 𝑖𝑡
5553, 54nffv 6916 . . . . . . . . . . . . . . 15 𝑖(𝐹𝑡)
56 nfcv 2905 . . . . . . . . . . . . . . 15 𝑖1
5755, 56nffv 6916 . . . . . . . . . . . . . 14 𝑖((𝐹𝑡)‘1)
58 nffvmpt1 6917 . . . . . . . . . . . . . 14 𝑖((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)
5957, 58nfeq 2919 . . . . . . . . . . . . 13 𝑖((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)
6048, 59nfim 1896 . . . . . . . . . . . 12 𝑖(𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
61 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 1 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘1))
62 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 1 → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
6361, 62eqeq12d 2753 . . . . . . . . . . . . 13 (𝑖 = 1 → (((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) ↔ ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)))
6463imbi2d 340 . . . . . . . . . . . 12 (𝑖 = 1 → ((𝑡𝑇 → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖)) ↔ (𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))))
65 ovex 7464 . . . . . . . . . . . . . . 15 (1...𝑀) ∈ V
6665mptex 7243 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V
6749fvmpt2 7027 . . . . . . . . . . . . . 14 ((𝑡𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V) → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
6866, 67mpan2 691 . . . . . . . . . . . . 13 (𝑡𝑇 → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
6968fveq1d 6908 . . . . . . . . . . . 12 (𝑡𝑇 → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
7060, 64, 69vtoclg1f 3570 . . . . . . . . . . 11 (1 ∈ (1...𝑀) → (𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)))
7147, 70syl 17 . . . . . . . . . 10 (𝜑 → (𝑡𝑇 → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1)))
7271imp 406 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1))
73 eqid 2737 . . . . . . . . . 10 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
74 fveq2 6906 . . . . . . . . . . 11 (𝑖 = 1 → (𝑈𝑖) = (𝑈‘1))
7574fveq1d 6908 . . . . . . . . . 10 (𝑖 = 1 → ((𝑈𝑖)‘𝑡) = ((𝑈‘1)‘𝑡))
7647adantr 480 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 1 ∈ (1...𝑀))
77 fmuldfeq.9 . . . . . . . . . . . . 13 (𝜑𝑈:(1...𝑀)⟶𝑌)
7877, 47ffvelcdmd 7105 . . . . . . . . . . . 12 (𝜑 → (𝑈‘1) ∈ 𝑌)
7978ancli 548 . . . . . . . . . . . 12 (𝜑 → (𝜑 ∧ (𝑈‘1) ∈ 𝑌))
80 eleq1 2829 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈‘1) → (𝑓𝑌 ↔ (𝑈‘1) ∈ 𝑌))
8180anbi2d 630 . . . . . . . . . . . . . 14 (𝑓 = (𝑈‘1) → ((𝜑𝑓𝑌) ↔ (𝜑 ∧ (𝑈‘1) ∈ 𝑌)))
82 feq1 6716 . . . . . . . . . . . . . 14 (𝑓 = (𝑈‘1) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘1):𝑇⟶ℝ))
8381, 82imbi12d 344 . . . . . . . . . . . . 13 (𝑓 = (𝑈‘1) → (((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ)))
84 fmuldfeq.10 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)
8584a1i 11 . . . . . . . . . . . . 13 (𝑓𝑌 → ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ))
8683, 85vtoclga 3577 . . . . . . . . . . . 12 ((𝑈‘1) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘1) ∈ 𝑌) → (𝑈‘1):𝑇⟶ℝ))
8778, 79, 86sylc 65 . . . . . . . . . . 11 (𝜑 → (𝑈‘1):𝑇⟶ℝ)
8887ffvelcdmda 7104 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝑈‘1)‘𝑡) ∈ ℝ)
8973, 75, 76, 88fvmptd3 7039 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘1) = ((𝑈‘1)‘𝑡))
9072, 89eqtrd 2777 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((𝑈‘1)‘𝑡))
91 seq1 14055 . . . . . . . . . 10 (1 ∈ ℤ → (seq1(𝑃, 𝑈)‘1) = (𝑈‘1))
9241, 91ax-mp 5 . . . . . . . . 9 (seq1(𝑃, 𝑈)‘1) = (𝑈‘1)
9392fveq1i 6907 . . . . . . . 8 ((seq1(𝑃, 𝑈)‘1)‘𝑡) = ((𝑈‘1)‘𝑡)
9490, 93eqtr4di 2795 . . . . . . 7 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘1) = ((seq1(𝑃, 𝑈)‘1)‘𝑡))
9543, 94eqtr2id 2790 . . . . . 6 ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))
96953adant3 1133 . . . . 5 ((𝜑𝑡𝑇 ∧ 1 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘1)‘𝑡) = (seq1( · , (𝐹𝑡))‘1))
97 simp31 1210 . . . . . . 7 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝜑)
98 simp1 1137 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ ℕ)
99 simp33 1212 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → (𝑛 + 1) ∈ (1...𝑀))
10098, 99jca 511 . . . . . . . . 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)))
101 elnnuz 12922 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
102101biimpi 216 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
103102anim1i 615 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑀)) → (𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)))
104 peano2fzr 13577 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑀)) → 𝑛 ∈ (1...𝑀))
105100, 103, 1043syl 18 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑛 ∈ (1...𝑀))
106 simp32 1211 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ ((𝜑𝑡𝑇𝑛 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)) ∧ (𝜑𝑡𝑇 ∧ (𝑛 + 1) ∈ (1...𝑀))) → 𝑡𝑇)
107 simp2 1138 . . . . . . . . 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)))
10897, 106, 105, 107mp3and 1466 . . . . . . . 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))
109105, 99, 1083jca 1129 . . . . . . 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)))
110 nfv 1914 . . . . . . . . 9 𝑓𝜑
111 nfv 1914 . . . . . . . . . 10 𝑓 𝑛 ∈ (1...𝑀)
112 nfv 1914 . . . . . . . . . 10 𝑓(𝑛 + 1) ∈ (1...𝑀)
113 nfcv 2905 . . . . . . . . . . . . . 14 𝑓1
114 fmuldfeq.3 . . . . . . . . . . . . . . 15 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
115 nfmpo1 7513 . . . . . . . . . . . . . . 15 𝑓(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
116114, 115nfcxfr 2903 . . . . . . . . . . . . . 14 𝑓𝑃
117 nfcv 2905 . . . . . . . . . . . . . 14 𝑓𝑈
118113, 116, 117nfseq 14052 . . . . . . . . . . . . 13 𝑓seq1(𝑃, 𝑈)
119 nfcv 2905 . . . . . . . . . . . . 13 𝑓𝑛
120118, 119nffv 6916 . . . . . . . . . . . 12 𝑓(seq1(𝑃, 𝑈)‘𝑛)
121 nfcv 2905 . . . . . . . . . . . 12 𝑓𝑡
122120, 121nffv 6916 . . . . . . . . . . 11 𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
123 nfcv 2905 . . . . . . . . . . 11 𝑓(seq1( · , (𝐹𝑡))‘𝑛)
124122, 123nfeq 2919 . . . . . . . . . 10 𝑓((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)
125111, 112, 124nf3an 1901 . . . . . . . . 9 𝑓(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
126110, 125nfan 1899 . . . . . . . 8 𝑓(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
127 nfv 1914 . . . . . . . . 9 𝑔𝜑
128 nfv 1914 . . . . . . . . . 10 𝑔 𝑛 ∈ (1...𝑀)
129 nfv 1914 . . . . . . . . . 10 𝑔(𝑛 + 1) ∈ (1...𝑀)
130 nfcv 2905 . . . . . . . . . . . . . 14 𝑔1
131 nfmpo2 7514 . . . . . . . . . . . . . . 15 𝑔(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
132114, 131nfcxfr 2903 . . . . . . . . . . . . . 14 𝑔𝑃
133 nfcv 2905 . . . . . . . . . . . . . 14 𝑔𝑈
134130, 132, 133nfseq 14052 . . . . . . . . . . . . 13 𝑔seq1(𝑃, 𝑈)
135 nfcv 2905 . . . . . . . . . . . . 13 𝑔𝑛
136134, 135nffv 6916 . . . . . . . . . . . 12 𝑔(seq1(𝑃, 𝑈)‘𝑛)
137 nfcv 2905 . . . . . . . . . . . 12 𝑔𝑡
138136, 137nffv 6916 . . . . . . . . . . 11 𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡)
139 nfcv 2905 . . . . . . . . . . 11 𝑔(seq1( · , (𝐹𝑡))‘𝑛)
140138, 139nfeq 2919 . . . . . . . . . 10 𝑔((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)
141128, 129, 140nf3an 1901 . . . . . . . . 9 𝑔(𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))
142127, 141nfan 1899 . . . . . . . 8 𝑔(𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛)))
143 fmuldfeq.2 . . . . . . . 8 𝑡𝑌
144 fmuldfeq.7 . . . . . . . . 9 (𝜑𝑇 ∈ V)
145144adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑇 ∈ V)
14677adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑈:(1...𝑀)⟶𝑌)
147 fmuldfeq.11 . . . . . . . . 9 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
1481473adant1r 1178 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) ∧ 𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
149 simpr1 1195 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → 𝑛 ∈ (1...𝑀))
150 simpr2 1196 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) → (𝑛 + 1) ∈ (1...𝑀))
151 simpr3 1197 . . . . . . . 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))
15284adantlr 715 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ (1...𝑀) ∧ (𝑛 + 1) ∈ (1...𝑀) ∧ ((seq1(𝑃, 𝑈)‘𝑛)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑛))) ∧ 𝑓𝑌) → 𝑓:𝑇⟶ℝ)
153126, 142, 143, 114, 49, 145, 146, 148, 149, 150, 151, 152fmuldfeqlem1 45597 . . . . . . 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)))
15497, 109, 106, 153syl21anc 838 . . . . . 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)))
1551543exp 1120 . . . . 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)))))
15619, 26, 33, 40, 96, 155nnind 12284 . . . 4 (𝑀 ∈ ℕ → ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀)))
15712, 156mpcom 38 . . 3 ((𝜑𝑡𝑇𝑀 ∈ (1...𝑀)) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
15811, 157mpd3an3 1464 . 2 ((𝜑𝑡𝑇) → ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
159 fmuldfeq.4 . . . 4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
160159fveq1i 6907 . . 3 (𝑋𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡)
161160a1i 11 . 2 ((𝜑𝑡𝑇) → (𝑋𝑡) = ((seq1(𝑃, 𝑈)‘𝑀)‘𝑡))
162 simpr 484 . . 3 ((𝜑𝑡𝑇) → 𝑡𝑇)
163 elnnuz 12922 . . . . . 6 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ‘1))
1642, 163sylib 218 . . . . 5 (𝜑𝑀 ∈ (ℤ‘1))
165164adantr 480 . . . 4 ((𝜑𝑡𝑇) → 𝑀 ∈ (ℤ‘1))
166 fmuldfeq.1 . . . . . . . 8 𝑖𝜑
167166, 48nfan 1899 . . . . . . 7 𝑖(𝜑𝑡𝑇)
168 nfv 1914 . . . . . . 7 𝑖 𝑘 ∈ (1...𝑀)
169167, 168nfan 1899 . . . . . 6 𝑖((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀))
170 nfcv 2905 . . . . . . . 8 𝑖𝑘
17155, 170nffv 6916 . . . . . . 7 𝑖((𝐹𝑡)‘𝑘)
172171nfel1 2922 . . . . . 6 𝑖((𝐹𝑡)‘𝑘) ∈ ℝ
173169, 172nfim 1896 . . . . 5 𝑖(((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
174 eleq1 2829 . . . . . . 7 (𝑖 = 𝑘 → (𝑖 ∈ (1...𝑀) ↔ 𝑘 ∈ (1...𝑀)))
175174anbi2d 630 . . . . . 6 (𝑖 = 𝑘 → (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀))))
176 fveq2 6906 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘𝑘))
177176eleq1d 2826 . . . . . 6 (𝑖 = 𝑘 → (((𝐹𝑡)‘𝑖) ∈ ℝ ↔ ((𝐹𝑡)‘𝑘) ∈ ℝ))
178175, 177imbi12d 344 . . . . 5 (𝑖 = 𝑘 → ((((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ∈ ℝ) ↔ (((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)))
17969ad2antlr 727 . . . . . 6 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
180 simpr 484 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
18177ffvelcdmda 7104 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
182 simpl 482 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
183182, 181jca 511 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈𝑖) ∈ 𝑌))
184 eleq1 2829 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → (𝑓𝑌 ↔ (𝑈𝑖) ∈ 𝑌))
185184anbi2d 630 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝑌) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝑌)))
186 feq1 6716 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
187185, 186imbi12d 344 . . . . . . . . . . . 12 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝑌) → (𝑈𝑖):𝑇⟶ℝ)))
188187, 85vtoclga 3577 . . . . . . . . . . 11 ((𝑈𝑖) ∈ 𝑌 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝑌) → (𝑈𝑖):𝑇⟶ℝ))
189181, 183, 188sylc 65 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
190189adantlr 715 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
191 simplr 769 . . . . . . . . 9 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
192190, 191ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
19373fvmpt2 7027 . . . . . . . 8 ((𝑖 ∈ (1...𝑀) ∧ ((𝑈𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
194180, 192, 193syl2anc 584 . . . . . . 7 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
195194, 192eqeltrd 2841 . . . . . 6 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) ∈ ℝ)
196179, 195eqeltrd 2841 . . . . 5 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ∈ ℝ)
197173, 178, 196chvarfv 2240 . . . 4 (((𝜑𝑡𝑇) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
198 remulcl 11240 . . . . 5 ((𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑘 · 𝑏) ∈ ℝ)
199198adantl 481 . . . 4 (((𝜑𝑡𝑇) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑘 · 𝑏) ∈ ℝ)
200165, 197, 199seqcl 14063 . . 3 ((𝜑𝑡𝑇) → (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ)
201 fmuldfeq.6 . . . 4 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
202201fvmpt2 7027 . . 3 ((𝑡𝑇 ∧ (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
203162, 200, 202syl2anc 584 . 2 ((𝜑𝑡𝑇) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
204158, 161, 2033eqtr4d 2787 1 ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wnf 1783  wcel 2108  wnfc 2890  Vcvv 3480   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  cr 11154  1c1 11156   + caddc 11158   · cmul 11160  cle 11296  cn 12266  cz 12613  cuz 12878  ...cfz 13547  seqcseq 14042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-seq 14043
This theorem is referenced by:  stoweidlem42  46057  stoweidlem48  46063
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