| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > constrextdg2 | Structured version Visualization version GIF version | ||
| Description: Any step (𝐶‘𝑁) of the construction of constructible numbers is contained in the last field of a tower of quadratic field extensions starting with ℚ. See Theorem 7.11 of [Stewart] p. 97. (Contributed by Thierry Arnoux, 19-Oct-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}) |
| constrextdg2.1 | ⊢ 𝐸 = (ℂfld ↾s 𝑒) |
| constrextdg2.2 | ⊢ 𝐹 = (ℂfld ↾s 𝑓) |
| constrextdg2.l | ⊢ < = {〈𝑓, 𝑒〉 ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)} |
| constrextdg2.n | ⊢ (𝜑 → 𝑁 ∈ ω) |
| Ref | Expression |
|---|---|
| constrextdg2 | ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | constrextdg2.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ω) | |
| 2 | fveq2 6881 | . . . . . 6 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅)) | |
| 3 | 2 | sseq1d 3967 | . . . . 5 ⊢ (𝑚 = ∅ → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘∅) ⊆ (lastS‘𝑣))) |
| 4 | 3 | anbi2d 641 | . . . 4 ⊢ (𝑚 = ∅ → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)))) |
| 5 | 4 | rexbidv 3187 | . . 3 ⊢ (𝑚 = ∅ → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)))) |
| 6 | fveq2 6881 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛)) | |
| 7 | 6 | sseq1d 3967 | . . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑛) ⊆ (lastS‘𝑣))) |
| 8 | 7 | anbi2d 641 | . . . . 5 ⊢ (𝑚 = 𝑛 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)))) |
| 9 | 8 | rexbidv 3187 | . . . 4 ⊢ (𝑚 = 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)))) |
| 10 | fveq1 6880 | . . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0)) | |
| 11 | 10 | eqeq1d 2763 | . . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ)) |
| 12 | fveq2 6881 | . . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢)) | |
| 13 | 12 | sseq2d 3968 | . . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝐶‘𝑛) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑛) ⊆ (lastS‘𝑢))) |
| 14 | 11, 13 | anbi12d 643 | . . . . 5 ⊢ (𝑣 = 𝑢 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)) ↔ ((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)))) |
| 15 | 14 | cbvrexvw 3242 | . . . 4 ⊢ (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))) |
| 16 | 9, 15 | bitrdi 290 | . . 3 ⊢ (𝑚 = 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)))) |
| 17 | fveq2 6881 | . . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛)) | |
| 18 | 17 | sseq1d 3967 | . . . . 5 ⊢ (𝑚 = suc 𝑛 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))) |
| 19 | 18 | anbi2d 641 | . . . 4 ⊢ (𝑚 = suc 𝑛 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))) |
| 20 | 19 | rexbidv 3187 | . . 3 ⊢ (𝑚 = suc 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))) |
| 21 | fveq2 6881 | . . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁)) | |
| 22 | 21 | sseq1d 3967 | . . . . 5 ⊢ (𝑚 = 𝑁 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑁) ⊆ (lastS‘𝑣))) |
| 23 | 22 | anbi2d 641 | . . . 4 ⊢ (𝑚 = 𝑁 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))) |
| 24 | 23 | rexbidv 3187 | . . 3 ⊢ (𝑚 = 𝑁 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))) |
| 25 | fveq1 6880 | . . . . . . 7 ⊢ (𝑣 = 〈“ℚ”〉 → (𝑣‘0) = (〈“ℚ”〉‘0)) | |
| 26 | 25 | eqeq1d 2763 | . . . . . 6 ⊢ (𝑣 = 〈“ℚ”〉 → ((𝑣‘0) = ℚ ↔ (〈“ℚ”〉‘0) = ℚ)) |
| 27 | fveq2 6881 | . . . . . . 7 ⊢ (𝑣 = 〈“ℚ”〉 → (lastS‘𝑣) = (lastS‘〈“ℚ”〉)) | |
| 28 | 27 | sseq2d 3968 | . . . . . 6 ⊢ (𝑣 = 〈“ℚ”〉 → ((𝐶‘∅) ⊆ (lastS‘𝑣) ↔ (𝐶‘∅) ⊆ (lastS‘〈“ℚ”〉))) |
| 29 | 26, 28 | anbi12d 643 | . . . . 5 ⊢ (𝑣 = 〈“ℚ”〉 → (((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)) ↔ ((〈“ℚ”〉‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘〈“ℚ”〉)))) |
| 30 | cndrng 21530 | . . . . . . . 8 ⊢ ℂfld ∈ DivRing | |
| 31 | qsubdrg 21548 | . . . . . . . . 9 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing) | |
| 32 | 31 | simpli 488 | . . . . . . . 8 ⊢ ℚ ∈ (SubRing‘ℂfld) |
| 33 | 31 | simpri 490 | . . . . . . . 8 ⊢ (ℂfld ↾s ℚ) ∈ DivRing |
| 34 | issdrg 20870 | . . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)) | |
| 35 | 30, 32, 33, 34 | mpbir3an 1358 | . . . . . . 7 ⊢ ℚ ∈ (SubDRing‘ℂfld) |
| 36 | 35 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℚ ∈ (SubDRing‘ℂfld)) |
| 37 | 36 | s1chn 18675 | . . . . 5 ⊢ (⊤ → 〈“ℚ”〉 ∈ ( < Chain (SubDRing‘ℂfld))) |
| 38 | s1fv 14648 | . . . . . . 7 ⊢ (ℚ ∈ (SubDRing‘ℂfld) → (〈“ℚ”〉‘0) = ℚ) | |
| 39 | 36, 38 | syl 18 | . . . . . 6 ⊢ (⊤ → (〈“ℚ”〉‘0) = ℚ) |
| 40 | 0z 12601 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
| 41 | 1z 12623 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
| 42 | prssi 4786 | . . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ) | |
| 43 | 40, 41, 42 | mp2an 704 | . . . . . . . 8 ⊢ {0, 1} ⊆ ℤ |
| 44 | zssq 12979 | . . . . . . . 8 ⊢ ℤ ⊆ ℚ | |
| 45 | 43, 44 | sstri 3945 | . . . . . . 7 ⊢ {0, 1} ⊆ ℚ |
| 46 | constr0.1 | . . . . . . . 8 ⊢ 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}) | |
| 47 | 46 | constr0 34093 | . . . . . . 7 ⊢ (𝐶‘∅) = {0, 1} |
| 48 | lsws1 14649 | . . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂfld) → (lastS‘〈“ℚ”〉) = ℚ) | |
| 49 | 35, 48 | ax-mp 5 | . . . . . . 7 ⊢ (lastS‘〈“ℚ”〉) = ℚ |
| 50 | 45, 47, 49 | 3sstr4i 3987 | . . . . . 6 ⊢ (𝐶‘∅) ⊆ (lastS‘〈“ℚ”〉) |
| 51 | 39, 50 | jctir 529 | . . . . 5 ⊢ (⊤ → ((〈“ℚ”〉‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘〈“ℚ”〉))) |
| 52 | 29, 37, 51 | rspcedvdw 3583 | . . . 4 ⊢ (⊤ → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣))) |
| 53 | 52 | mptru 1575 | . . 3 ⊢ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)) |
| 54 | constrextdg2.1 | . . . . . 6 ⊢ 𝐸 = (ℂfld ↾s 𝑒) | |
| 55 | constrextdg2.2 | . . . . . 6 ⊢ 𝐹 = (ℂfld ↾s 𝑓) | |
| 56 | constrextdg2.l | . . . . . 6 ⊢ < = {〈𝑓, 𝑒〉 ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)} | |
| 57 | simplll 786 | . . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → 𝑛 ∈ ω) | |
| 58 | simpllr 787 | . . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) | |
| 59 | simplr 780 | . . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → (𝑢‘0) = ℚ) | |
| 60 | simpr 489 | . . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → (𝐶‘𝑛) ⊆ (lastS‘𝑢)) | |
| 61 | 46, 54, 55, 56, 57, 58, 59, 60 | constrextdg2lem 34104 | . . . . 5 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))) |
| 62 | 61 | anasss 471 | . . . 4 ⊢ (((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))) |
| 63 | 62 | rexlimdva2 3166 | . . 3 ⊢ (𝑛 ∈ ω → (∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))) |
| 64 | 5, 16, 20, 24, 53, 63 | finds 7892 | . 2 ⊢ (𝑁 ∈ ω → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣))) |
| 65 | 1, 64 | syl 18 | 1 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 400 ∨ w3o 1100 ∧ w3a 1101 = wceq 1568 ⊤wtru 1569 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3087 {crab 3414 Vcvv 3453 ⊆ wss 3904 ∅c0 4285 {cpr 4590 class class class wbr 5108 {copab 5172 ↦ cmpt 5191 suc csuc 6362 ‘cfv 6536 (class class class)co 7410 ωcom 7861 reccrdg 8395 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 + caddc 11102 · cmul 11104 − cmin 11440 2c2 12294 ℤcz 12590 ℚcq 12971 lastSclsw 14599 〈“cs1 14633 ∗ccj 15147 ℑcim 15149 abscabs 15285 ↾s cress 17289 Chain cchn 18660 SubRingcsubrg 20653 DivRingcdr 20812 SubDRingcsdrg 20868 ℂfldccnfld 21501 /FldExtcfldext 33994 [:]cextdg 33996 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-reg 9553 ax-inf2 9609 ax-ac2 10446 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178 ax-mulf 11179 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-rpss 7720 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-er 8693 df-ec 8695 df-qs 8699 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-sup 9401 df-inf 9402 df-oi 9471 df-r1 9735 df-rank 9736 df-dju 9886 df-card 9924 df-acn 9927 df-ac 10099 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-xnn0 12577 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-ico 13377 df-fz 13535 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-word 14551 df-lsw 14600 df-concat 14608 df-s1 14634 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ocomp 17330 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-0g 17493 df-gsum 17494 df-prds 17499 df-pws 17501 df-imas 17561 df-qus 17562 df-mre 17637 df-mrc 17638 df-mri 17639 df-acs 17640 df-proset 18349 df-drs 18350 df-poset 18368 df-ipo 18583 df-chn 18661 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-subg 19188 df-nsg 19189 df-eqg 19190 df-ghm 19283 df-gim 19328 df-cntz 19386 df-oppg 19415 df-lsm 19705 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-srg 20268 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-irred 20440 df-invr 20469 df-dvr 20482 df-rhm 20553 df-nzr 20595 df-subrng 20630 df-subrg 20654 df-rlreg 20778 df-domn 20779 df-idom 20780 df-drng 20814 df-field 20815 df-sdrg 20869 df-lmod 20962 df-lss 21032 df-lsp 21072 df-lmhm 21122 df-lmim 21123 df-lmic 21124 df-lbs 21175 df-lvec 21203 df-sra 21273 df-rgmod 21274 df-lidl 21311 df-rsp 21312 df-2idl 21368 df-lpidl 21469 df-lpir 21470 df-pid 21484 df-cnfld 21502 df-dsmm 21861 df-frlm 21876 df-uvc 21912 df-lindf 21935 df-linds 21936 df-assa 21982 df-asp 21983 df-ascl 21984 df-psr 22038 df-mvr 22039 df-mpl 22040 df-opsr 22042 df-evls 22204 df-evl 22205 df-psr1 22319 df-vr1 22320 df-ply1 22321 df-coe1 22322 df-evls1 22454 df-evl1 22455 df-mdeg 26191 df-deg1 26192 df-mon1 26267 df-uc1p 26268 df-q1p 26269 df-r1p 26270 df-ig1p 26271 df-fldgen 33598 df-mxidl 33709 df-dim 33956 df-fldext 33997 df-extdg 33998 df-irng 34040 df-minply 34056 |
| This theorem is referenced by: constrext2chnlem 34106 |
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