Theorem fimgmcyclem 42543

Description: Lemma for fimgmcyc 42544. (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 3290	. . . . . 6 ⊢ (∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
3		eqcom 2744	. . . . . . . 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 3129	. . . . . 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 5147	. . . . . . . 8 ⊢ (𝑜 = 𝑟 → (𝑝 < 𝑜 ↔ 𝑝 < 𝑟))
8		oveq1 7438	. . . . . . . . 9 ⊢ (𝑜 = 𝑟 → (𝑜 · 𝐴) = (𝑟 · 𝐴))
9	8	eqeq1d 2739	. . . . . . . 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 3179	. . . . . 6 ⊢ (𝑜 = 𝑟 → (∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴))))
12	11	cbvrexvw 3238	. . . . 5 ⊢ (∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
13		breq1 5146	. . . . . . . 8 ⊢ (𝑜 = 𝑝 → (𝑜 < 𝑟 ↔ 𝑝 < 𝑟))
14		oveq1 7438	. . . . . . . . 9 ⊢ (𝑜 = 𝑝 → (𝑜 · 𝐴) = (𝑝 · 𝐴))
15	14	eqeq1d 2739	. . . . . . . 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 3179	. . . . . 6 ⊢ (𝑜 = 𝑝 → (∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴))))
18	17	cbvrexvw 3238	. . . . 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 5146	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝑞 < 𝑜 ↔ 𝑝 < 𝑜))
21		oveq1 7438	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 · 𝐴) = (𝑝 · 𝐴))
22	21	eqeq2d 2748	. . . . . . 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 3238	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
25	24	rexbii 3094	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
26		breq2 5147	. . . . . . 7 ⊢ (𝑞 = 𝑟 → (𝑜 < 𝑞 ↔ 𝑜 < 𝑟))
27		oveq1 7438	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 · 𝐴) = (𝑟 · 𝐴))
28	27	eqeq2d 2748	. . . . . . 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 3238	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
31	30	rexbii 3094	. . . 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 12281	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑜 ∈ ℝ)
37		simpr 484	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℕ)
38	37	nnred 12281	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℝ)
39	36, 38	lttri2d 11400	. . . . . . 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 1011	. . . . . 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 3218	. . . 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 3122	. . . . 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 3094	. . . 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 3122	. . . 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 961	1 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070   class class class wbr 5143  (class class class)co 7431   < clt 11295  ℕcn 12266

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-i2m1 11223  ax-1ne0 11224  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-ltxr 11300  df-nn 12267

This theorem is referenced by:  fimgmcyc  42544