Theorem fimgmcyclem 43019

Description: Lemma for fimgmcyc 43020. (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 485	. 2 ⊢ ((𝜑 ∧ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))) → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
2		rexcom 3268	. . . . . 6 ⊢ (∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
3		eqcom 2746	. . . . . . . 8 ⊢ ((𝑟 · 𝐴) = (𝑝 · 𝐴) ↔ (𝑝 · 𝐴) = (𝑟 · 𝐴))
4	3	anbi2i 629	. . . . . . 7 ⊢ ((𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
5	4	2rexbii 3115	. . . . . 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 220	. . . . 5 ⊢ (∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)) → ∃𝑝 ∈ ℕ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
7		breq2 5076	. . . . . . . 8 ⊢ (𝑜 = 𝑟 → (𝑝 < 𝑜 ↔ 𝑝 < 𝑟))
8		oveq1 7363	. . . . . . . . 9 ⊢ (𝑜 = 𝑟 → (𝑜 · 𝐴) = (𝑟 · 𝐴))
9	8	eqeq1d 2741	. . . . . . . 8 ⊢ (𝑜 = 𝑟 → ((𝑜 · 𝐴) = (𝑝 · 𝐴) ↔ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
10	7, 9	anbi12d 638	. . . . . . 7 ⊢ (𝑜 = 𝑟 → ((𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴))))
11	10	rexbidv 3163	. . . . . 6 ⊢ (𝑜 = 𝑟 → (∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴))))
12	11	cbvrexvw 3218	. . . . 5 ⊢ (∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) ↔ ∃𝑟 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑟 · 𝐴) = (𝑝 · 𝐴)))
13		breq1 5075	. . . . . . . 8 ⊢ (𝑜 = 𝑝 → (𝑜 < 𝑟 ↔ 𝑝 < 𝑟))
14		oveq1 7363	. . . . . . . . 9 ⊢ (𝑜 = 𝑝 → (𝑜 · 𝐴) = (𝑝 · 𝐴))
15	14	eqeq1d 2741	. . . . . . . 8 ⊢ (𝑜 = 𝑝 → ((𝑜 · 𝐴) = (𝑟 · 𝐴) ↔ (𝑝 · 𝐴) = (𝑟 · 𝐴)))
16	13, 15	anbi12d 638	. . . . . . 7 ⊢ (𝑜 = 𝑝 → ((𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴))))
17	16	rexbidv 3163	. . . . . 6 ⊢ (𝑜 = 𝑝 → (∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑝 < 𝑟 ∧ (𝑝 · 𝐴) = (𝑟 · 𝐴))))
18	17	cbvrexvw 3218	. . . . 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 293	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)) → ∃𝑜 ∈ ℕ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
20		breq1 5075	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝑞 < 𝑜 ↔ 𝑝 < 𝑜))
21		oveq1 7363	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 · 𝐴) = (𝑝 · 𝐴))
22	21	eqeq2d 2750	. . . . . . 7 ⊢ (𝑞 = 𝑝 → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
23	20, 22	anbi12d 638	. . . . . 6 ⊢ (𝑞 = 𝑝 → ((𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴))))
24	23	cbvrexvw 3218	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
25	24	rexbii 3086	. . . 4 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑝 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑝 · 𝐴)))
26		breq2 5076	. . . . . . 7 ⊢ (𝑞 = 𝑟 → (𝑜 < 𝑞 ↔ 𝑜 < 𝑟))
27		oveq1 7363	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 · 𝐴) = (𝑟 · 𝐴))
28	27	eqeq2d 2750	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
29	26, 28	anbi12d 638	. . . . . 6 ⊢ (𝑞 = 𝑟 → ((𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴))))
30	29	cbvrexvw 3218	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑟 ∈ ℕ (𝑜 < 𝑟 ∧ (𝑜 · 𝐴) = (𝑟 · 𝐴)))
31	30	rexbii 3086	. . . 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 293	. . 3 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑞 < 𝑜 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
33	32	adantl 482	. 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 483	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑜 ∈ ℕ)
36	35	nnred 12180	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑜 ∈ ℝ)
37		simpr 485	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℕ)
38	37	nnred 12180	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → 𝑞 ∈ ℝ)
39	36, 38	lttri2d 11276	. . . . . . 7 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑜 ≠ 𝑞 ↔ (𝑜 < 𝑞 ∨ 𝑞 < 𝑜)))
40	39	anbi1d 637	. . . . . 6 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑜 < 𝑞 ∨ 𝑞 < 𝑜) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
41		andir 1016	. . . . . 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 288	. . . . 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 3200	. . . 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 3107	. . . . 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 3086	. . . 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 3107	. . . 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 298	. . 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 219	. 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 966	1 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063   class class class wbr 5072  (class class class)co 7356   < clt 11170  ℕcn 12165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-i2m1 11097  ax-1ne0 11098  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-ltxr 11175  df-nn 12166

This theorem is referenced by:  fimgmcyc  43020