| 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 6885 | . . . . . 6 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅)) | |
| 3 | 2 | sseq1d 3976 | . . . . 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 3196 | . . 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 6885 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛)) | |
| 7 | 6 | sseq1d 3976 | . . . . . 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 3196 | . . . 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 6884 | . . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0)) | |
| 11 | 10 | eqeq1d 2772 | . . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ)) |
| 12 | fveq2 6885 | . . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢)) | |
| 13 | 12 | sseq2d 3977 | . . . . . 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 3251 | . . . 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 6885 | . . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛)) | |
| 18 | 17 | sseq1d 3976 | . . . . 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 3196 | . . 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 6885 | . . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁)) | |
| 22 | 21 | sseq1d 3976 | . . . . 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 3196 | . . 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 6884 | . . . . . . 7 ⊢ (𝑣 = 〈“ℚ”〉 → (𝑣‘0) = (〈“ℚ”〉‘0)) | |
| 26 | 25 | eqeq1d 2772 | . . . . . 6 ⊢ (𝑣 = 〈“ℚ”〉 → ((𝑣‘0) = ℚ ↔ (〈“ℚ”〉‘0) = ℚ)) |
| 27 | fveq2 6885 | . . . . . . 7 ⊢ (𝑣 = 〈“ℚ”〉 → (lastS‘𝑣) = (lastS‘〈“ℚ”〉)) | |
| 28 | 27 | sseq2d 3977 | . . . . . 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 21534 | . . . . . . . 8 ⊢ ℂfld ∈ DivRing | |
| 31 | qsubdrg 21552 | . . . . . . . . 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 20874 | . . . . . . . 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 18679 | . . . . 5 ⊢ (⊤ → 〈“ℚ”〉 ∈ ( < Chain (SubDRing‘ℂfld))) |
| 38 | s1fv 14651 | . . . . . . 7 ⊢ (ℚ ∈ (SubDRing‘ℂfld) → (〈“ℚ”〉‘0) = ℚ) | |
| 39 | 36, 38 | syl 18 | . . . . . 6 ⊢ (⊤ → (〈“ℚ”〉‘0) = ℚ) |
| 40 | 0z 12605 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
| 41 | 1z 12627 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
| 42 | prssi 4792 | . . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ) | |
| 43 | 40, 41, 42 | mp2an 704 | . . . . . . . 8 ⊢ {0, 1} ⊆ ℤ |
| 44 | zssq 12983 | . . . . . . . 8 ⊢ ℤ ⊆ ℚ | |
| 45 | 43, 44 | sstri 3954 | . . . . . . 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 34097 | . . . . . . 7 ⊢ (𝐶‘∅) = {0, 1} |
| 48 | lsws1 14652 | . . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂfld) → (lastS‘〈“ℚ”〉) = ℚ) | |
| 49 | 35, 48 | ax-mp 5 | . . . . . . 7 ⊢ (lastS‘〈“ℚ”〉) = ℚ |
| 50 | 45, 47, 49 | 3sstr4i 3996 | . . . . . 6 ⊢ (𝐶‘∅) ⊆ (lastS‘〈“ℚ”〉) |
| 51 | 39, 50 | jctir 529 | . . . . 5 ⊢ (⊤ → ((〈“ℚ”〉‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘〈“ℚ”〉))) |
| 52 | 29, 37, 51 | rspcedvdw 3592 | . . . 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 34108 | . . . . 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 3175 | . . 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 7896 | . 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 2150 ≠ wne 2965 ∃wrex 3096 {crab 3423 Vcvv 3462 ⊆ wss 3913 ∅c0 4294 {cpr 4596 class class class wbr 5114 {copab 5178 ↦ cmpt 5197 suc csuc 6366 ‘cfv 6540 (class class class)co 7414 ωcom 7865 reccrdg 8399 ℂcc 11101 ℝcr 11102 0cc0 11103 1c1 11104 + caddc 11106 · cmul 11108 − cmin 11444 2c2 12298 ℤcz 12594 ℚcq 12975 lastSclsw 14602 〈“cs1 14636 ∗ccj 15150 ℑcim 15152 abscabs 15288 ↾s cress 17293 Chain cchn 18664 SubRingcsubrg 20657 DivRingcdr 20816 SubDRingcsdrg 20872 ℂfldccnfld 21505 /FldExtcfldext 33998 [:]cextdg 34000 |
| 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 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-reg 9557 ax-inf2 9613 ax-ac2 10450 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180 ax-pre-sup 11181 ax-addf 11182 ax-mulf 11183 |
| 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 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-se 5619 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7678 df-ofr 7679 df-rpss 7724 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8160 df-tpos 8225 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-er 8697 df-ec 8699 df-qs 8703 df-map 8829 df-pm 8830 df-ixp 8899 df-en 8947 df-dom 8948 df-sdom 8949 df-fin 8950 df-fsupp 9325 df-sup 9405 df-inf 9406 df-oi 9475 df-r1 9739 df-rank 9740 df-dju 9890 df-card 9928 df-acn 9931 df-ac 10103 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-div 11875 df-nn 12237 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12508 df-xnn0 12581 df-z 12595 df-dec 12715 df-uz 12866 df-q 12976 df-rp 13020 df-ico 13381 df-fz 13539 df-fzo 13686 df-seq 14041 df-exp 14101 df-hash 14370 df-word 14554 df-lsw 14603 df-concat 14611 df-s1 14637 df-cj 15153 df-re 15154 df-im 15155 df-sqrt 15289 df-abs 15290 df-struct 17210 df-sets 17227 df-slot 17245 df-ndx 17257 df-base 17273 df-ress 17294 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ocomp 17334 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-0g 17497 df-gsum 17498 df-prds 17503 df-pws 17505 df-imas 17565 df-qus 17566 df-mre 17641 df-mrc 17642 df-mri 17643 df-acs 17644 df-proset 18353 df-drs 18354 df-poset 18372 df-ipo 18587 df-chn 18665 df-mgm 18701 df-sgrp 18780 df-mnd 18796 df-mhm 18844 df-submnd 18845 df-grp 19006 df-minusg 19007 df-sbg 19008 df-mulg 19137 df-subg 19192 df-nsg 19193 df-eqg 19194 df-ghm 19287 df-gim 19332 df-cntz 19390 df-oppg 19419 df-lsm 19709 df-cmn 19855 df-abl 19856 df-mgp 20220 df-rng 20234 df-ur 20267 df-srg 20272 df-ring 20320 df-cring 20321 df-oppr 20422 df-dvdsr 20442 df-unit 20443 df-irred 20444 df-invr 20473 df-dvr 20486 df-rhm 20557 df-nzr 20599 df-subrng 20634 df-subrg 20658 df-rlreg 20782 df-domn 20783 df-idom 20784 df-drng 20818 df-field 20819 df-sdrg 20873 df-lmod 20966 df-lss 21036 df-lsp 21076 df-lmhm 21126 df-lmim 21127 df-lmic 21128 df-lbs 21179 df-lvec 21207 df-sra 21277 df-rgmod 21278 df-lidl 21315 df-rsp 21316 df-2idl 21372 df-lpidl 21473 df-lpir 21474 df-pid 21488 df-cnfld 21506 df-dsmm 21865 df-frlm 21880 df-uvc 21916 df-lindf 21939 df-linds 21940 df-assa 21986 df-asp 21987 df-ascl 21988 df-psr 22042 df-mvr 22043 df-mpl 22044 df-opsr 22046 df-evls 22208 df-evl 22209 df-psr1 22323 df-vr1 22324 df-ply1 22325 df-coe1 22326 df-evls1 22458 df-evl1 22459 df-mdeg 26195 df-deg1 26196 df-mon1 26271 df-uc1p 26272 df-q1p 26273 df-r1p 26274 df-ig1p 26275 df-fldgen 33602 df-mxidl 33713 df-dim 33960 df-fldext 34001 df-extdg 34002 df-irng 34044 df-minply 34060 |
| This theorem is referenced by: constrext2chnlem 34110 |
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