Theorem fimgmcyclem 43359

Description: Lemma for fimgmcyc 43360. (Contributed by SN, 7-Jul-2025.)

Hypothesis

Ref	Expression
fimgmcyclem.s	⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))

Assertion

Ref	Expression
fimgmcyclem	⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))

Distinct variable groups:   · ,𝑜,𝑞   𝐴,𝑜,𝑞   𝜑,𝑜,𝑞

Proof of Theorem fimgmcyclem

Dummy variables 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. 2 ⊢ ((𝜑 ∧ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))) → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
2		rexcom 3296	. . . . . 6 ⊢ (∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
3		eqcom 2772	. . . . . . . 8 ⊢ ((𝑟 · 𝐴) = (𝑝 · 𝐴) ↔ (𝑝 · 𝐴) = (𝑟 · 𝐴))
4	3	anbi2i 635	. . . . . . 7 ⊢ ((𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
5	4	2rexbii 3143	. . . . . 6 ⊢ (∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
6	2, 5	sylbb 222	. . . . 5 ⊢ (∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) → ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
7		breq2 5115	. . . . . . . 8 ⊢ (𝑜 = 𝑟 → (𝑝 < 𝑜 ↔ 𝑝 < 𝑟))
8		oveq1 7426	. . . . . . . . 9 ⊢ (𝑜 = 𝑟 → (𝑜 · 𝐴) = (𝑟 · 𝐴))
9	8	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑜 = 𝑟 → ((𝑜 · 𝐴) = (𝑝 · 𝐴) ↔ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
10	7, 9	anbi12d 644	. . . . . . 7 ⊢ (𝑜 = 𝑟 → ((𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴))))
11	10	rexbidv 3191	. . . . . 6 ⊢ (𝑜 = 𝑟 → (∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴))))
12	11	cbvrexvw 3246	. . . . 5 ⊢ (∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
13		breq1 5114	. . . . . . . 8 ⊢ (𝑜 = 𝑝 → (𝑜 < 𝑟 ↔ 𝑝 < 𝑟))
14		oveq1 7426	. . . . . . . . 9 ⊢ (𝑜 = 𝑝 → (𝑜 · 𝐴) = (𝑝 · 𝐴))
15	14	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑜 = 𝑝 → ((𝑜 · 𝐴) = (𝑟 · 𝐴) ↔ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
16	13, 15	anbi12d 644	. . . . . . 7 ⊢ (𝑜 = 𝑝 → ((𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴))))
17	16	rexbidv 3191	. . . . . 6 ⊢ (𝑜 = 𝑝 → (∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴))))
18	17	cbvrexvw 3246	. . . . 5 ⊢ (∃𝑜 ∈ ℕ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
19	6, 12, 18	3imtr4i 295	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) → ∃𝑜 ∈ ℕ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
20		breq1 5114	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝑞 < 𝑜 ↔ 𝑝 < 𝑜))
21		oveq1 7426	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 · 𝐴) = (𝑝 · 𝐴))
22	21	eqeq2d 2776	. . . . . . 7 ⊢ (𝑞 = 𝑝 → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
23	20, 22	anbi12d 644	. . . . . 6 ⊢ (𝑞 = 𝑝 → ((𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴))))
24	23	cbvrexvw 3246	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
25	24	rexbii 3114	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
26		breq2 5115	. . . . . . 7 ⊢ (𝑞 = 𝑟 → (𝑜 < 𝑞 ↔ 𝑜 < 𝑟))
27		oveq1 7426	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 · 𝐴) = (𝑟 · 𝐴))
28	27	eqeq2d 2776	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
29	26, 28	anbi12d 644	. . . . . 6 ⊢ (𝑞 = 𝑟 → ((𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴))))
30	29	cbvrexvw 3246	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
31	30	rexbii 3114	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
32	19, 25, 31	3imtr4i 295	. . 3 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
33	32	adantl 487	. 2 ⊢ ((𝜑 ∧ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))) → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
34		fimgmcyclem.s	. . 3 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
35		simpl 488	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑜 ∈ ℕ)
36	35	nnred 12263	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑜 ∈ ℝ)
37		simpr 490	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℕ)
38	37	nnred 12263	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℝ)
39	36, 38	lttri2d 11364	. . . . . . 7 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑜 ≠ 𝑞 ↔ (𝑜 < 𝑞 ∨ 𝑞 < 𝑜)))
40	39	anbi1d 643	. . . . . 6 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑜 < 𝑞 ∨ 𝑞 < 𝑜) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
41		andir 1026	. . . . . 6 ⊢ (((𝑜 < 𝑞 ∨ 𝑞 < 𝑜) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
42	40, 41	bitrdi 290	. . . . 5 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
43	42	2rexbiia 3228	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ((𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
44		r19.43 3135	. . . . 5 ⊢ (∃𝑞 ∈ ℕ ((𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))) ↔ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
45	44	rexbii 3114	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ((𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))) ↔ ∃𝑜 ∈ ℕ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
46		r19.43 3135	. . . 4 ⊢ (∃𝑜 ∈ ℕ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))) ↔ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
47	43, 45, 46	3bitri 300	. . 3 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
48	34, 47	sylib 221	. 2 ⊢ (𝜑 → (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ∨ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
49	1, 33, 48	mpjaodan 973	1 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091   class class class wbr 5111  (class class class)co 7419   < clt 11258  ℕcn 12248

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-i2m1 11183  ax-1ne0 11184  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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-ov 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11260  df-mnf 11261  df-ltxr 11263  df-nn 12249

This theorem is used by:  fimgmcyc  43360