Theorem fimgmcyclem 42788

Description: Lemma for fimgmcyc 42789. (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 3265	. . . . . 6 ⊢ (∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
3		eqcom 2743	. . . . . . . 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 3112	. . . . . 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 5102	. . . . . . . 8 ⊢ (𝑜 = 𝑟 → (𝑝 < 𝑜 ↔ 𝑝 < 𝑟))
8		oveq1 7365	. . . . . . . . 9 ⊢ (𝑜 = 𝑟 → (𝑜 · 𝐴) = (𝑟 · 𝐴))
9	8	eqeq1d 2738	. . . . . . . 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 3160	. . . . . 6 ⊢ (𝑜 = 𝑟 → (∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴))))
12	11	cbvrexvw 3215	. . . . 5 ⊢ (∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
13		breq1 5101	. . . . . . . 8 ⊢ (𝑜 = 𝑝 → (𝑜 < 𝑟 ↔ 𝑝 < 𝑟))
14		oveq1 7365	. . . . . . . . 9 ⊢ (𝑜 = 𝑝 → (𝑜 · 𝐴) = (𝑝 · 𝐴))
15	14	eqeq1d 2738	. . . . . . . 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 3160	. . . . . 6 ⊢ (𝑜 = 𝑝 → (∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴))))
18	17	cbvrexvw 3215	. . . . 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 5101	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝑞 < 𝑜 ↔ 𝑝 < 𝑜))
21		oveq1 7365	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 · 𝐴) = (𝑝 · 𝐴))
22	21	eqeq2d 2747	. . . . . . 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 3215	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
25	24	rexbii 3083	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
26		breq2 5102	. . . . . . 7 ⊢ (𝑞 = 𝑟 → (𝑜 < 𝑞 ↔ 𝑜 < 𝑟))
27		oveq1 7365	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 · 𝐴) = (𝑟 · 𝐴))
28	27	eqeq2d 2747	. . . . . . 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 3215	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
31	30	rexbii 3083	. . . 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 12160	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑜 ∈ ℝ)
37		simpr 484	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℕ)
38	37	nnred 12160	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℝ)
39	36, 38	lttri2d 11272	. . . . . . 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 3197	. . . 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 3104	. . . . 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 3083	. . . 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 3104	. . . 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 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060   class class class wbr 5098  (class class class)co 7358   < clt 11166  ℕcn 12145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-i2m1 11094  ax-1ne0 11095  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-ltxr 11171  df-nn 12146

This theorem is referenced by:  fimgmcyc  42789