Theorem fimgmcyclem 42506

Description: Lemma for fimgmcyc 42507. (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 484	. 2 ⊢ ((𝜑 ∧ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))) → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
2		rexcom 3258	. . . . . 6 ⊢ (∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
3		eqcom 2736	. . . . . . . 8 ⊢ ((𝑟 · 𝐴) = (𝑝 · 𝐴) ↔ (𝑝 · 𝐴) = (𝑟 · 𝐴))
4	3	anbi2i 623	. . . . . . 7 ⊢ ((𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
5	4	2rexbii 3105	. . . . . 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 219	. . . . 5 ⊢ (∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) → ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
7		breq2 5096	. . . . . . . 8 ⊢ (𝑜 = 𝑟 → (𝑝 < 𝑜 ↔ 𝑝 < 𝑟))
8		oveq1 7356	. . . . . . . . 9 ⊢ (𝑜 = 𝑟 → (𝑜 · 𝐴) = (𝑟 · 𝐴))
9	8	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑜 = 𝑟 → ((𝑜 · 𝐴) = (𝑝 · 𝐴) ↔ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
10	7, 9	anbi12d 632	. . . . . . 7 ⊢ (𝑜 = 𝑟 → ((𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴))))
11	10	rexbidv 3153	. . . . . 6 ⊢ (𝑜 = 𝑟 → (∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴))))
12	11	cbvrexvw 3208	. . . . 5 ⊢ (∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
13		breq1 5095	. . . . . . . 8 ⊢ (𝑜 = 𝑝 → (𝑜 < 𝑟 ↔ 𝑝 < 𝑟))
14		oveq1 7356	. . . . . . . . 9 ⊢ (𝑜 = 𝑝 → (𝑜 · 𝐴) = (𝑝 · 𝐴))
15	14	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑜 = 𝑝 → ((𝑜 · 𝐴) = (𝑟 · 𝐴) ↔ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
16	13, 15	anbi12d 632	. . . . . . 7 ⊢ (𝑜 = 𝑝 → ((𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴))))
17	16	rexbidv 3153	. . . . . 6 ⊢ (𝑜 = 𝑝 → (∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴))))
18	17	cbvrexvw 3208	. . . . 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 292	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) → ∃𝑜 ∈ ℕ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
20		breq1 5095	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝑞 < 𝑜 ↔ 𝑝 < 𝑜))
21		oveq1 7356	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 · 𝐴) = (𝑝 · 𝐴))
22	21	eqeq2d 2740	. . . . . . 7 ⊢ (𝑞 = 𝑝 → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
23	20, 22	anbi12d 632	. . . . . 6 ⊢ (𝑞 = 𝑝 → ((𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴))))
24	23	cbvrexvw 3208	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
25	24	rexbii 3076	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
26		breq2 5096	. . . . . . 7 ⊢ (𝑞 = 𝑟 → (𝑜 < 𝑞 ↔ 𝑜 < 𝑟))
27		oveq1 7356	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 · 𝐴) = (𝑟 · 𝐴))
28	27	eqeq2d 2740	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
29	26, 28	anbi12d 632	. . . . . 6 ⊢ (𝑞 = 𝑟 → ((𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴))))
30	29	cbvrexvw 3208	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
31	30	rexbii 3076	. . . 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 292	. . 3 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
33	32	adantl 481	. 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 482	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑜 ∈ ℕ)
36	35	nnred 12143	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑜 ∈ ℝ)
37		simpr 484	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℕ)
38	37	nnred 12143	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℝ)
39	36, 38	lttri2d 11255	. . . . . . 7 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑜 ≠ 𝑞 ↔ (𝑜 < 𝑞 ∨ 𝑞 < 𝑜)))
40	39	anbi1d 631	. . . . . 6 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑜 < 𝑞 ∨ 𝑞 < 𝑜) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
41		andir 1010	. . . . . 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 287	. . . . 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 3190	. . . 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 3097	. . . . 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 3076	. . . 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 3097	. . . 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 297	. . 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 218	. 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 960	1 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   class class class wbr 5092  (class class class)co 7349   < clt 11149  ℕcn 12128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-i2m1 11077  ax-1ne0 11078  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-ltxr 11154  df-nn 12129

This theorem is referenced by:  fimgmcyc  42507