Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  constrsslem Structured version   Visualization version   GIF version

Theorem constrsslem 33731
Description: Lemma for constrss 33733. This lemma requires the additional condition that 0 is the constructible number; that condition is removed in constrss 33733. (Proposed by Saveliy Skresanov, 23-JUn-2025.) (Contributed by Thierry Arnoux, 25-Jun-2025.)
Hypotheses
Ref Expression
constr0.1 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎𝑠𝑏𝑠𝑐𝑠𝑑𝑠𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑𝑐))) ∧ (ℑ‘((∗‘(𝑏𝑎)) · (𝑑𝑐))) ≠ 0) ∨ ∃𝑎𝑠𝑏𝑠𝑐𝑠𝑒𝑠𝑓𝑠𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ∨ ∃𝑎𝑠𝑏𝑠𝑐𝑠𝑑𝑠𝑒𝑠𝑓𝑠 (𝑎𝑑 ∧ (abs‘(𝑥𝑎)) = (abs‘(𝑏𝑐)) ∧ (abs‘(𝑥𝑑)) = (abs‘(𝑒𝑓))))}), {0, 1})
constrsscn.1 (𝜑𝑁 ∈ On)
constrsslem.1 (𝜑 → 0 ∈ (𝐶𝑁))
Assertion
Ref Expression
constrsslem (𝜑 → (𝐶𝑁) ⊆ (𝐶‘suc 𝑁))
Distinct variable groups:   𝐶,𝑎,𝑠,𝑥,𝑏,𝑐   𝐶,𝑑,𝑠,𝑥   𝐶,𝑒,𝑠,𝑥,𝑓   𝑠,𝑟,𝑥   𝑡,𝑠,𝑥,𝐶   𝑎,𝑏,𝑐,𝑒,𝑓,𝑡,𝑁   𝑁,𝑑,𝑠,𝑥   𝜑,𝑎,𝑏,𝑐,𝑒,𝑓,𝑠,𝑡,𝑥
Allowed substitution hints:   𝜑(𝑟,𝑑)   𝐶(𝑟)   𝑁(𝑟)

Proof of Theorem constrsslem
StepHypRef Expression
1 constr0.1 . . . . . 6 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎𝑠𝑏𝑠𝑐𝑠𝑑𝑠𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑𝑐))) ∧ (ℑ‘((∗‘(𝑏𝑎)) · (𝑑𝑐))) ≠ 0) ∨ ∃𝑎𝑠𝑏𝑠𝑐𝑠𝑒𝑠𝑓𝑠𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ∨ ∃𝑎𝑠𝑏𝑠𝑐𝑠𝑑𝑠𝑒𝑠𝑓𝑠 (𝑎𝑑 ∧ (abs‘(𝑥𝑎)) = (abs‘(𝑏𝑐)) ∧ (abs‘(𝑥𝑑)) = (abs‘(𝑒𝑓))))}), {0, 1})
2 constrsscn.1 . . . . . 6 (𝜑𝑁 ∈ On)
31, 2constrsscn 33730 . . . . 5 (𝜑 → (𝐶𝑁) ⊆ ℂ)
43sselda 4008 . . . 4 ((𝜑𝑥 ∈ (𝐶𝑁)) → 𝑥 ∈ ℂ)
5 simpr 484 . . . . . 6 ((𝜑𝑥 ∈ (𝐶𝑁)) → 𝑥 ∈ (𝐶𝑁))
6 id 22 . . . . . . . . . . . . 13 (𝑎 = 𝑥𝑎 = 𝑥)
7 oveq2 7456 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (𝑏𝑎) = (𝑏𝑥))
87oveq2d 7464 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑡 · (𝑏𝑎)) = (𝑡 · (𝑏𝑥)))
96, 8oveq12d 7466 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝑎 + (𝑡 · (𝑏𝑎))) = (𝑥 + (𝑡 · (𝑏𝑥))))
109eqeq2d 2751 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ↔ 𝑥 = (𝑥 + (𝑡 · (𝑏𝑥)))))
1110anbi1d 630 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
1211rexbidv 3185 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
13122rexbidv 3228 . . . . . . . 8 (𝑎 = 𝑥 → (∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
14132rexbidv 3228 . . . . . . 7 (𝑎 = 𝑥 → (∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
1514adantl 481 . . . . . 6 (((𝜑𝑥 ∈ (𝐶𝑁)) ∧ 𝑎 = 𝑥) → (∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
16 constrsslem.1 . . . . . . . 8 (𝜑 → 0 ∈ (𝐶𝑁))
1716adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶𝑁)) → 0 ∈ (𝐶𝑁))
18 oveq1 7455 . . . . . . . . . . . . . 14 (𝑏 = 0 → (𝑏𝑥) = (0 − 𝑥))
1918oveq2d 7464 . . . . . . . . . . . . 13 (𝑏 = 0 → (𝑡 · (𝑏𝑥)) = (𝑡 · (0 − 𝑥)))
2019oveq2d 7464 . . . . . . . . . . . 12 (𝑏 = 0 → (𝑥 + (𝑡 · (𝑏𝑥))) = (𝑥 + (𝑡 · (0 − 𝑥))))
2120eqeq2d 2751 . . . . . . . . . . 11 (𝑏 = 0 → (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ↔ 𝑥 = (𝑥 + (𝑡 · (0 − 𝑥)))))
2221anbi1d 630 . . . . . . . . . 10 (𝑏 = 0 → ((𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
23222rexbidv 3228 . . . . . . . . 9 (𝑏 = 0 → (∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
24232rexbidv 3228 . . . . . . . 8 (𝑏 = 0 → (∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
2524adantl 481 . . . . . . 7 (((𝜑𝑥 ∈ (𝐶𝑁)) ∧ 𝑏 = 0) → (∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)))))
26 oveq2 7456 . . . . . . . . . . . . . 14 (𝑐 = 0 → (𝑥𝑐) = (𝑥 − 0))
2726fveq2d 6924 . . . . . . . . . . . . 13 (𝑐 = 0 → (abs‘(𝑥𝑐)) = (abs‘(𝑥 − 0)))
2827eqeq1d 2742 . . . . . . . . . . . 12 (𝑐 = 0 → ((abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓)) ↔ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓))))
2928anbi2d 629 . . . . . . . . . . 11 (𝑐 = 0 → ((𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓)))))
3029rexbidv 3185 . . . . . . . . . 10 (𝑐 = 0 → (∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓)))))
31302rexbidv 3228 . . . . . . . . 9 (𝑐 = 0 → (∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓)))))
3231adantl 481 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐶𝑁)) ∧ 𝑐 = 0) → (∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ↔ ∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓)))))
33 oveq1 7455 . . . . . . . . . . . . . 14 (𝑒 = 𝑥 → (𝑒𝑓) = (𝑥𝑓))
3433fveq2d 6924 . . . . . . . . . . . . 13 (𝑒 = 𝑥 → (abs‘(𝑒𝑓)) = (abs‘(𝑥𝑓)))
3534eqeq2d 2751 . . . . . . . . . . . 12 (𝑒 = 𝑥 → ((abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓)) ↔ (abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓))))
3635anbi2d 629 . . . . . . . . . . 11 (𝑒 = 𝑥 → ((𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓))) ↔ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓)))))
37362rexbidv 3228 . . . . . . . . . 10 (𝑒 = 𝑥 → (∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓))) ↔ ∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓)))))
3837adantl 481 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐶𝑁)) ∧ 𝑒 = 𝑥) → (∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓))) ↔ ∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓)))))
39 oveq2 7456 . . . . . . . . . . . . . . 15 (𝑓 = 0 → (𝑥𝑓) = (𝑥 − 0))
4039fveq2d 6924 . . . . . . . . . . . . . 14 (𝑓 = 0 → (abs‘(𝑥𝑓)) = (abs‘(𝑥 − 0)))
4140eqeq2d 2751 . . . . . . . . . . . . 13 (𝑓 = 0 → ((abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓)) ↔ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0))))
4241anbi2d 629 . . . . . . . . . . . 12 (𝑓 = 0 → ((𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓))) ↔ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0)))))
4342rexbidv 3185 . . . . . . . . . . 11 (𝑓 = 0 → (∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓))) ↔ ∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0)))))
4443adantl 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝐶𝑁)) ∧ 𝑓 = 0) → (∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓))) ↔ ∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0)))))
45 0red 11293 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐶𝑁)) → 0 ∈ ℝ)
46 oveq1 7455 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑡 · (0 − 𝑥)) = (0 · (0 − 𝑥)))
4746oveq2d 7464 . . . . . . . . . . . . . 14 (𝑡 = 0 → (𝑥 + (𝑡 · (0 − 𝑥))) = (𝑥 + (0 · (0 − 𝑥))))
4847eqeq2d 2751 . . . . . . . . . . . . 13 (𝑡 = 0 → (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ↔ 𝑥 = (𝑥 + (0 · (0 − 𝑥)))))
4948anbi1d 630 . . . . . . . . . . . 12 (𝑡 = 0 → ((𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0))) ↔ (𝑥 = (𝑥 + (0 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0)))))
5049adantl 481 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝐶𝑁)) ∧ 𝑡 = 0) → ((𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0))) ↔ (𝑥 = (𝑥 + (0 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0)))))
51 0cnd 11283 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐶𝑁)) → 0 ∈ ℂ)
5251, 4subcld 11647 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐶𝑁)) → (0 − 𝑥) ∈ ℂ)
5352mul02d 11488 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐶𝑁)) → (0 · (0 − 𝑥)) = 0)
5453oveq2d 7464 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐶𝑁)) → (𝑥 + (0 · (0 − 𝑥))) = (𝑥 + 0))
554addridd 11490 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐶𝑁)) → (𝑥 + 0) = 𝑥)
5654, 55eqtr2d 2781 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐶𝑁)) → 𝑥 = (𝑥 + (0 · (0 − 𝑥))))
57 eqidd 2741 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐶𝑁)) → (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0)))
5856, 57jca 511 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐶𝑁)) → (𝑥 = (𝑥 + (0 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0))))
5945, 50, 58rspcedvd 3637 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐶𝑁)) → ∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥 − 0))))
6017, 44, 59rspcedvd 3637 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐶𝑁)) → ∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑥𝑓))))
615, 38, 60rspcedvd 3637 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐶𝑁)) → ∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥 − 0)) = (abs‘(𝑒𝑓))))
6217, 32, 61rspcedvd 3637 . . . . . . 7 ((𝜑𝑥 ∈ (𝐶𝑁)) → ∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (0 − 𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))))
6317, 25, 62rspcedvd 3637 . . . . . 6 ((𝜑𝑥 ∈ (𝐶𝑁)) → ∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑥 + (𝑡 · (𝑏𝑥))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))))
645, 15, 63rspcedvd 3637 . . . . 5 ((𝜑𝑥 ∈ (𝐶𝑁)) → ∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))))
65643mix2d 1337 . . . 4 ((𝜑𝑥 ∈ (𝐶𝑁)) → (∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑑 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑𝑐))) ∧ (ℑ‘((∗‘(𝑏𝑎)) · (𝑑𝑐))) ≠ 0) ∨ ∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ∨ ∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑑 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)(𝑎𝑑 ∧ (abs‘(𝑥𝑎)) = (abs‘(𝑏𝑐)) ∧ (abs‘(𝑥𝑑)) = (abs‘(𝑒𝑓)))))
66 eqid 2740 . . . . . 6 (𝐶𝑁) = (𝐶𝑁)
671, 2, 66constrsuc 33728 . . . . 5 (𝜑 → (𝑥 ∈ (𝐶‘suc 𝑁) ↔ (𝑥 ∈ ℂ ∧ (∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑑 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑𝑐))) ∧ (ℑ‘((∗‘(𝑏𝑎)) · (𝑑𝑐))) ≠ 0) ∨ ∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ∨ ∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑑 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)(𝑎𝑑 ∧ (abs‘(𝑥𝑎)) = (abs‘(𝑏𝑐)) ∧ (abs‘(𝑥𝑑)) = (abs‘(𝑒𝑓)))))))
6867adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝐶𝑁)) → (𝑥 ∈ (𝐶‘suc 𝑁) ↔ (𝑥 ∈ ℂ ∧ (∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑑 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑𝑐))) ∧ (ℑ‘((∗‘(𝑏𝑎)) · (𝑑𝑐))) ≠ 0) ∨ ∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏𝑎))) ∧ (abs‘(𝑥𝑐)) = (abs‘(𝑒𝑓))) ∨ ∃𝑎 ∈ (𝐶𝑁)∃𝑏 ∈ (𝐶𝑁)∃𝑐 ∈ (𝐶𝑁)∃𝑑 ∈ (𝐶𝑁)∃𝑒 ∈ (𝐶𝑁)∃𝑓 ∈ (𝐶𝑁)(𝑎𝑑 ∧ (abs‘(𝑥𝑎)) = (abs‘(𝑏𝑐)) ∧ (abs‘(𝑥𝑑)) = (abs‘(𝑒𝑓)))))))
694, 65, 68mpbir2and 712 . . 3 ((𝜑𝑥 ∈ (𝐶𝑁)) → 𝑥 ∈ (𝐶‘suc 𝑁))
7069ex 412 . 2 (𝜑 → (𝑥 ∈ (𝐶𝑁) → 𝑥 ∈ (𝐶‘suc 𝑁)))
7170ssrdv 4014 1 (𝜑 → (𝐶𝑁) ⊆ (𝐶‘suc 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3o 1086  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wrex 3076  {crab 3443  Vcvv 3488  wss 3976  {cpr 4650  cmpt 5249  Oncon0 6395  suc csuc 6397  cfv 6573  (class class class)co 7448  reccrdg 8465  cc 11182  cr 11183  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189  cmin 11520  ccj 15145  cim 15147  abscabs 15283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-ltxr 11329  df-sub 11522
This theorem is referenced by:  constr01  33732  constrss  33733
  Copyright terms: Public domain W3C validator