Theorem symgmatr01lem 21908

Description: Lemma for symgmatr01 21909. (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 764	. . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝐾 ∈ 𝑁)
2		eqeq1 2740	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘 = 𝐾 ↔ 𝐾 = 𝐾))
3		fveq2 6825	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑄‘𝑘) = (𝑄‘𝐾))
4	3	eqeq1d 2738	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝑄‘𝑘) = 𝐿 ↔ (𝑄‘𝐾) = 𝐿))
5	4	ifbid 4496	. . . . . 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 7355	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘𝑀(𝑄‘𝑘)) = (𝐾𝑀(𝑄‘𝐾)))
8	2, 5, 7	ifbieq12d 4501	. . . . 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 2738	. . . 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 482	. . 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 2737	. . . . 5 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝐾 = 𝐾)
12	11	iftrued 4481	. . . 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 3908	. . . . . . 7 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}))
14		ianor 979	. . . . . . . . . 10 ⊢ (¬ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿) ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))
15		fveq1 6824	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑄 → (𝑞‘𝐾) = (𝑄‘𝐾))
16	15	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑄 → ((𝑞‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = 𝐿))
17	16	elrab 3634	. . . . . . . . . 10 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿))
18	14, 17	xchnxbir 332	. . . . . . . . 9 ⊢ (¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))
19		pm2.21 123	. . . . . . . . . 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 854	. . . . . . . . 9 ⊢ ((¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿) → (𝑄 ∈ 𝑃 → ¬ (𝑄‘𝐾) = 𝐿))
22	18, 21	sylbi 216	. . . . . . . 8 ⊢ (¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} → (𝑄 ∈ 𝑃 → ¬ (𝑄‘𝐾) = 𝐿))
23	22	impcom 408	. . . . . . 7 ⊢ ((𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ¬ (𝑄‘𝐾) = 𝐿)
24	13, 23	sylbi 216	. . . . . 6 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ¬ (𝑄‘𝐾) = 𝐿)
25	24	adantl 482	. . . . 5 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → ¬ (𝑄‘𝐾) = 𝐿)
26	25	iffalsed 4484	. . . 4 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵) = 𝐵)
27	12, 26	eqtrd 2776	. . 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 3572	. 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 413	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 205   ∧ wa 396   ∨ wo 844   = wceq 1540   ∈ wcel 2105  ∃wrex 3070  {crab 3403   ∖ cdif 3895  ifcif 4473  ‘cfv 6479  (class class class)co 7337  Basecbs 17009  SymGrpcsymg 19070

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487  df-ov 7340

This theorem is referenced by:  symgmatr01  21909