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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > constr01 | Structured version Visualization version GIF version |
Description: 0 and 1 are in all steps of the construction of constructible points. (Contributed by Thierry Arnoux, 25-Jun-2025.) |
Ref | Expression |
---|---|
constr0.1 | ⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1}) |
constrsscn.1 | ⊢ (𝜑 → 𝑁 ∈ On) |
Ref | Expression |
---|---|
constr01 | ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | constrsscn.1 | . 2 ⊢ (𝜑 → 𝑁 ∈ On) | |
2 | fveq2 6920 | . . . 4 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅)) | |
3 | 2 | sseq2d 4041 | . . 3 ⊢ (𝑚 = ∅ → ({0, 1} ⊆ (𝐶‘𝑚) ↔ {0, 1} ⊆ (𝐶‘∅))) |
4 | fveq2 6920 | . . . 4 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛)) | |
5 | 4 | sseq2d 4041 | . . 3 ⊢ (𝑚 = 𝑛 → ({0, 1} ⊆ (𝐶‘𝑚) ↔ {0, 1} ⊆ (𝐶‘𝑛))) |
6 | fveq2 6920 | . . . 4 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛)) | |
7 | 6 | sseq2d 4041 | . . 3 ⊢ (𝑚 = suc 𝑛 → ({0, 1} ⊆ (𝐶‘𝑚) ↔ {0, 1} ⊆ (𝐶‘suc 𝑛))) |
8 | fveq2 6920 | . . . 4 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁)) | |
9 | 8 | sseq2d 4041 | . . 3 ⊢ (𝑚 = 𝑁 → ({0, 1} ⊆ (𝐶‘𝑚) ↔ {0, 1} ⊆ (𝐶‘𝑁))) |
10 | constr0.1 | . . . . 5 ⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1}) | |
11 | 10 | constr0 33727 | . . . 4 ⊢ (𝐶‘∅) = {0, 1} |
12 | 11 | eqimss2i 4070 | . . 3 ⊢ {0, 1} ⊆ (𝐶‘∅) |
13 | simpr 484 | . . . . 5 ⊢ ((𝑛 ∈ On ∧ {0, 1} ⊆ (𝐶‘𝑛)) → {0, 1} ⊆ (𝐶‘𝑛)) | |
14 | simpl 482 | . . . . . 6 ⊢ ((𝑛 ∈ On ∧ {0, 1} ⊆ (𝐶‘𝑛)) → 𝑛 ∈ On) | |
15 | c0ex 11284 | . . . . . . . . 9 ⊢ 0 ∈ V | |
16 | 15 | prid1 4787 | . . . . . . . 8 ⊢ 0 ∈ {0, 1} |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ ((𝑛 ∈ On ∧ {0, 1} ⊆ (𝐶‘𝑛)) → 0 ∈ {0, 1}) |
18 | 13, 17 | sseldd 4009 | . . . . . 6 ⊢ ((𝑛 ∈ On ∧ {0, 1} ⊆ (𝐶‘𝑛)) → 0 ∈ (𝐶‘𝑛)) |
19 | 10, 14, 18 | constrsslem 33731 | . . . . 5 ⊢ ((𝑛 ∈ On ∧ {0, 1} ⊆ (𝐶‘𝑛)) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛)) |
20 | 13, 19 | sstrd 4019 | . . . 4 ⊢ ((𝑛 ∈ On ∧ {0, 1} ⊆ (𝐶‘𝑛)) → {0, 1} ⊆ (𝐶‘suc 𝑛)) |
21 | 20 | ex 412 | . . 3 ⊢ (𝑛 ∈ On → ({0, 1} ⊆ (𝐶‘𝑛) → {0, 1} ⊆ (𝐶‘suc 𝑛))) |
22 | 0ellim 6458 | . . . . . 6 ⊢ (Lim 𝑚 → ∅ ∈ 𝑚) | |
23 | fveq2 6920 | . . . . . . . 8 ⊢ (𝑜 = ∅ → (𝐶‘𝑜) = (𝐶‘∅)) | |
24 | 23, 11 | eqtrdi 2796 | . . . . . . 7 ⊢ (𝑜 = ∅ → (𝐶‘𝑜) = {0, 1}) |
25 | 24 | ssiun2s 5071 | . . . . . 6 ⊢ (∅ ∈ 𝑚 → {0, 1} ⊆ ∪ 𝑜 ∈ 𝑚 (𝐶‘𝑜)) |
26 | 22, 25 | syl 17 | . . . . 5 ⊢ (Lim 𝑚 → {0, 1} ⊆ ∪ 𝑜 ∈ 𝑚 (𝐶‘𝑜)) |
27 | vex 3492 | . . . . . . 7 ⊢ 𝑚 ∈ V | |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (Lim 𝑚 → 𝑚 ∈ V) |
29 | id 22 | . . . . . 6 ⊢ (Lim 𝑚 → Lim 𝑚) | |
30 | 10, 28, 29 | constrlim 33729 | . . . . 5 ⊢ (Lim 𝑚 → (𝐶‘𝑚) = ∪ 𝑜 ∈ 𝑚 (𝐶‘𝑜)) |
31 | 26, 30 | sseqtrrd 4050 | . . . 4 ⊢ (Lim 𝑚 → {0, 1} ⊆ (𝐶‘𝑚)) |
32 | 31 | a1d 25 | . . 3 ⊢ (Lim 𝑚 → (∀𝑛 ∈ 𝑚 {0, 1} ⊆ (𝐶‘𝑛) → {0, 1} ⊆ (𝐶‘𝑚))) |
33 | 3, 5, 7, 9, 12, 21, 32 | tfinds 7897 | . 2 ⊢ (𝑁 ∈ On → {0, 1} ⊆ (𝐶‘𝑁)) |
34 | 1, 33 | syl 17 | 1 ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1086 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 Vcvv 3488 ⊆ wss 3976 ∅c0 4352 {cpr 4650 ∪ ciun 5015 ↦ cmpt 5249 Oncon0 6395 Lim wlim 6396 suc csuc 6397 ‘cfv 6573 (class class class)co 7448 reccrdg 8465 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 − cmin 11520 ∗ccj 15145 ℑcim 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: constrss 33733 constrelextdg2 33737 |
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