Theorem edomediv 33936

Description: Apply a Euclidean division in a Euclidean domain. (Contributed by Thierry Arnoux, 29-Oct-2026.)

Hypotheses

Ref	Expression
edomediv.b	⊢ 𝐵 = (Base‘𝐷)
edomediv.z	⊢ 0 = (0g‘𝐷)
edomediv.p	⊢ + = (+g‘𝐷)
edomediv.m	⊢ · = (.r‘𝐷)
edomediv.e	⊢ 𝐸 = (EuclF‘𝐷)
edomediv.d	⊢ (𝜑 → 𝐷 ∈ EDomn)
edomediv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
edomediv.y	⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))

Assertion

Ref	Expression
edomediv	⊢ (𝜑 → ∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌))))

Distinct variable groups:   0 ,𝑞,𝑟   𝐵,𝑞,𝑟   𝐷,𝑞,𝑟   𝐸,𝑞,𝑟   𝑋,𝑞,𝑟   𝑌,𝑞,𝑟   𝜑,𝑞,𝑟

Allowed substitution hints:   + (𝑟, 𝑞)   · (𝑟, 𝑞)

Proof of Theorem edomediv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2767	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑦 · 𝑞) + 𝑟) ↔ 𝑋 = ((𝑦 · 𝑞) + 𝑟)))
2	1	anbi1d 643	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦))) ↔ (𝑋 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦)))))
3	2	2rexbidv 3231	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑥 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦))) ↔ ∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑋 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦)))))
4		oveq1 7453	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 · 𝑞) = (𝑌 · 𝑞))
5	4	oveq1d 7461	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑦 · 𝑞) + 𝑟) = ((𝑌 · 𝑞) + 𝑟))
6	5	eqeq2d 2774	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = ((𝑦 · 𝑞) + 𝑟) ↔ 𝑋 = ((𝑌 · 𝑞) + 𝑟)))
7		fveq2 6911	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦) = ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌))
8	7	breq2d 5118	. . . . . 6 ⊢ (𝑦 = 𝑌 → (((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦) ↔ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌)))
9	8	orbi2d 929	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦)) ↔ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌))))
10	6, 9	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦))) ↔ (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌)))))
11	10	2rexbidv 3231	. . 3 ⊢ (𝑦 = 𝑌 → (∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑋 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦))) ↔ ∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌)))))
12		edomediv.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ EDomn)
13		edomediv.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
14		edomediv.z	. . . . . 6 ⊢ 0 = (0g‘𝐷)
15		edomediv.p	. . . . . 6 ⊢ + = (+g‘𝐷)
16		edomediv.m	. . . . . 6 ⊢ · = (.r‘𝐷)
17		edomediv.e	. . . . . . 7 ⊢ 𝐸 = (EuclF‘𝐷)
18	17	reseq1i 5978	. . . . . 6 ⊢ (𝐸 ↾ (𝐵 ∖ { 0 })) = ((EuclF‘𝐷) ↾ (𝐵 ∖ { 0 }))
19	13, 14, 15, 16, 18	isedom 33934	. . . . 5 ⊢ (𝐷 ∈ EDomn ↔ (𝐷 ∈ IDomn ∧ ((𝐸 ↾ (𝐵 ∖ { 0 })):(𝐵 ∖ { 0 })⟶ℕ0 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐵 ∖ { 0 })∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑥 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦))))))
20	12, 19	sylib 221	. . . 4 ⊢ (𝜑 → (𝐷 ∈ IDomn ∧ ((𝐸 ↾ (𝐵 ∖ { 0 })):(𝐵 ∖ { 0 })⟶ℕ0 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐵 ∖ { 0 })∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑥 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦))))))
21	20	simprrd 786	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐵 ∖ { 0 })∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑥 = ((𝑦 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑦))))
22		edomediv.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		edomediv.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))
24	3, 11, 21, 22, 23	rspc2dv 3594	. 2 ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌))))
25		biort 949	. . . . . . . 8 ⊢ (𝑟 = 0 → (𝑟 = 0 ↔ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌))))
26		biort 949	. . . . . . . 8 ⊢ (𝑟 = 0 → (𝑟 = 0 ↔ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌))))
27	25, 26	bitr3d 284	. . . . . . 7 ⊢ (𝑟 = 0 → ((𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌)) ↔ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌))))
28	27	adantl 487	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) ∧ 𝑟 = 0 ) → ((𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌)) ↔ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌))))
29		simplr 781	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) ∧ 𝑟 ≠ 0 ) → 𝑟 ∈ 𝐵)
30		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) ∧ 𝑟 ≠ 0 ) → 𝑟 ≠ 0 )
31	29, 30	eldifsnd 4750	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) ∧ 𝑟 ≠ 0 ) → 𝑟 ∈ (𝐵 ∖ { 0 }))
32	31	fvresd 6931	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) ∧ 𝑟 ≠ 0 ) → ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) = (𝐸‘𝑟))
33	23	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) → 𝑌 ∈ (𝐵 ∖ { 0 }))
34	33	fvresd 6931	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) → ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌) = (𝐸‘𝑌))
35	34	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) ∧ 𝑟 ≠ 0 ) → ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌) = (𝐸‘𝑌))
36	32, 35	breq12d 5119	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) ∧ 𝑟 ≠ 0 ) → (((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌) ↔ (𝐸‘𝑟) < (𝐸‘𝑌)))
37	36	orbi2d 929	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) ∧ 𝑟 ≠ 0 ) → ((𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌)) ↔ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌))))
38	28, 37	pm2.61dane 3046	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) → ((𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌)) ↔ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌))))
39	38	anbi2d 642	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐵) ∧ 𝑟 ∈ 𝐵) → ((𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌))) ↔ (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌)))))
40	39	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌))) ↔ (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌)))))
41	40	2rexbidva 3229	. 2 ⊢ (𝜑 → (∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑟) < ((𝐸 ↾ (𝐵 ∖ { 0 }))‘𝑌))) ↔ ∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌)))))
42	24, 41	mpbid 235	1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 ∃𝑟 ∈ 𝐵 (𝑋 = ((𝑌 · 𝑞) + 𝑟) ∧ (𝑟 = 0 ∨ (𝐸‘𝑟) < (𝐸‘𝑌))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ∖ cdif 4048  {csn 4588   class class class wbr 5106   ↾ cres 5658  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446   < clt 11373  ℕ₀cn0 12636  Basecbs 17419  +_gcplusg 17460  ._rcmulr 17461  0_gc0g 17642  IDomncidom 20978  EuclFceuf 33928  EDomncedom 33932

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7449  df-edom 33933

This theorem is used by:  edompidlem  33940