Theorem symgmatr01lem 20278

Description: Lemma for symgmatr01 20279. (Contributed by AV, 3-Jan-2019.)

Hypothesis

Ref	Expression
symgmatr01.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))

Assertion

Ref	Expression
symgmatr01lem	⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ 𝑁 if(𝑘 = 𝐾, if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄‘𝑘))) = 𝐵))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝑞,𝐿   𝑘,𝐾,𝑞   𝑘,𝑀   𝑘,𝑁   𝑃,𝑘,𝑞   𝑄,𝑘,𝑞

Allowed substitution hints:   𝐴(𝑞)   𝐵(𝑞)   𝑀(𝑞)   𝑁(𝑞)

Proof of Theorem symgmatr01lem

Step	Hyp	Ref	Expression
1		simpll 786	. . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝐾 ∈ 𝑁)
2		eqeq1 2614	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘 = 𝐾 ↔ 𝐾 = 𝐾))
3		fveq2 6103	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑄‘𝑘) = (𝑄‘𝐾))
4	3	eqeq1d 2612	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝑄‘𝑘) = 𝐿 ↔ (𝑄‘𝐾) = 𝐿))
5	4	ifbid 4058	. . . . . 6 ⊢ (𝑘 = 𝐾 → if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵) = if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵))
6		id 22	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
7	6, 3	oveq12d 6567	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘𝑀(𝑄‘𝑘)) = (𝐾𝑀(𝑄‘𝐾)))
8	2, 5, 7	ifbieq12d 4063	. . . . 5 ⊢ (𝑘 = 𝐾 → if(𝑘 = 𝐾, if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄‘𝑘))) = if(𝐾 = 𝐾, if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄‘𝐾))))
9	8	eqeq1d 2612	. . . 4 ⊢ (𝑘 = 𝐾 → (if(𝑘 = 𝐾, if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄‘𝑘))) = 𝐵 ↔ if(𝐾 = 𝐾, if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄‘𝐾))) = 𝐵))
10	9	adantl 481	. . 3 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) ∧ 𝑘 = 𝐾) → (if(𝑘 = 𝐾, if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄‘𝑘))) = 𝐵 ↔ if(𝐾 = 𝐾, if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄‘𝐾))) = 𝐵))
11		eqidd 2611	. . . . 5 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝐾 = 𝐾)
12	11	iftrued 4044	. . . 4 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → if(𝐾 = 𝐾, if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄‘𝐾))) = if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵))
13		eldif 3550	. . . . . . 7 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}))
14		ianor 508	. . . . . . . . . 10 ⊢ (¬ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿) ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))
15		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑄 → (𝑞‘𝐾) = (𝑄‘𝐾))
16	15	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑄 → ((𝑞‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = 𝐿))
17	16	elrab 3331	. . . . . . . . . 10 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿))
18	14, 17	xchnxbir 322	. . . . . . . . 9 ⊢ (¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))
19		pm2.21 119	. . . . . . . . . 10 ⊢ (¬ 𝑄 ∈ 𝑃 → (𝑄 ∈ 𝑃 → ¬ (𝑄‘𝐾) = 𝐿))
20		ax-1 6	. . . . . . . . . 10 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (𝑄 ∈ 𝑃 → ¬ (𝑄‘𝐾) = 𝐿))
21	19, 20	jaoi 393	. . . . . . . . 9 ⊢ ((¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿) → (𝑄 ∈ 𝑃 → ¬ (𝑄‘𝐾) = 𝐿))
22	18, 21	sylbi 206	. . . . . . . 8 ⊢ (¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} → (𝑄 ∈ 𝑃 → ¬ (𝑄‘𝐾) = 𝐿))
23	22	impcom 445	. . . . . . 7 ⊢ ((𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ¬ (𝑄‘𝐾) = 𝐿)
24	13, 23	sylbi 206	. . . . . 6 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ¬ (𝑄‘𝐾) = 𝐿)
25	24	adantl 481	. . . . 5 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → ¬ (𝑄‘𝐾) = 𝐿)
26	25	iffalsed 4047	. . . 4 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵) = 𝐵)
27	12, 26	eqtrd 2644	. . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → if(𝐾 = 𝐾, if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄‘𝐾))) = 𝐵)
28	1, 10, 27	rspcedvd 3289	. 2 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → ∃𝑘 ∈ 𝑁 if(𝑘 = 𝐾, if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄‘𝑘))) = 𝐵)
29	28	ex 449	1 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ 𝑁 if(𝑘 = 𝐾, if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄‘𝑘))) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900   ∖ cdif 3537  ifcif 4036  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  SymGrpcsymg 17620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  symgmatr01  20279