Theorem symgmatr01lem 22810

Description: Lemma for symgmatr01 22811. (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 778	. . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝐾 ∈ 𝑁)
2		eqeq1 2767	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘 = 𝐾 ↔ 𝐾 = 𝐾))
3		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑄‘𝑘) = (𝑄‘𝐾))
4	3	eqeq1d 2765	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝑄‘𝑘) = 𝐿 ↔ (𝑄‘𝐾) = 𝐿))
5	4	ifbid 4511	. . . . . 6 ⊢ (𝑘 = 𝐾 → if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵) = if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵))
6		id 23	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
7	6, 3	oveq12d 7428	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘𝑀(𝑄‘𝑘)) = (𝐾𝑀(𝑄‘𝐾)))
8	2, 5, 7	ifbieq12d 4516	. . . . 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 2765	. . . 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 486	. . 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 2764	. . . . 5 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝐾 = 𝐾)
12	11	iftrued 4495	. . . 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 3915	. . . . . . 7 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}))
14		ianor 997	. . . . . . . . . 10 ⊢ (¬ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿) ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))
15		fveq1 6880	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑄 → (𝑞‘𝐾) = (𝑄‘𝐾))
16	15	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑄 → ((𝑞‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = 𝐿))
17	16	elrab 3650	. . . . . . . . . 10 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿))
18	14, 17	xchnxbir 336	. . . . . . . . 9 ⊢ (¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))
19		pm2.21 124	. . . . . . . . . 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 870	. . . . . . . . 9 ⊢ ((¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿) → (𝑄 ∈ 𝑃 → ¬ (𝑄‘𝐾) = 𝐿))
22	18, 21	sylbi 220	. . . . . . . 8 ⊢ (¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} → (𝑄 ∈ 𝑃 → ¬ (𝑄‘𝐾) = 𝐿))
23	22	impcom 412	. . . . . . 7 ⊢ ((𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ¬ (𝑄‘𝐾) = 𝐿)
24	13, 23	sylbi 220	. . . . . 6 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ¬ (𝑄‘𝐾) = 𝐿)
25	24	adantl 486	. . . . 5 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → ¬ (𝑄‘𝐾) = 𝐿)
26	25	iffalsed 4498	. . . 4 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → if((𝑄‘𝐾) = 𝐿, 𝐴, 𝐵) = 𝐵)
27	12, 26	eqtrd 2798	. . 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 3583	. 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 417	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 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416   ∖ cdif 3902  ifcif 4487  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  SymGrpcsymg 19434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by:  symgmatr01  22811