Theorem addccan2nclem1 6264

Description: Lemma for addccan2nc 6266. Stratification helper theorem. (Contributed by Scott Fenton, 2-Aug-2019.)

Assertion

Ref	Expression
addccan2nclem1	⊢ (x( AddC ∘ ◡(1st ↾ (V × {n})))y ↔ y = (x +c n))

Distinct variable group:   x,n,y

Proof of Theorem addccan2nclem1

Dummy variables p q z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brco 4884	. . 3 ⊢ (x( AddC ∘ ◡(1st ↾ (V × {n})))y ↔ ∃z(x◡(1st ↾ (V × {n}))z ∧ z AddC y))
2		brcnv 4893	. . . . . 6 ⊢ (x◡(1st ↾ (V × {n}))z ↔ z(1st ↾ (V × {n}))x)
3		brres 4950	. . . . . 6 ⊢ (z(1st ↾ (V × {n}))x ↔ (z1st x ∧ z ∈ (V × {n})))
4		ancom 437	. . . . . . . . 9 ⊢ ((z1st x ∧ z ∈ (V × {n})) ↔ (z ∈ (V × {n}) ∧ z1st x))
5		elxp2 4803	. . . . . . . . . . 11 ⊢ (z ∈ (V × {n}) ↔ ∃p ∈ V ∃q ∈ {n}z = ⟨p, q⟩)
6		rexv 2874	. . . . . . . . . . 11 ⊢ (∃p ∈ V ∃q ∈ {n}z = ⟨p, q⟩ ↔ ∃p∃q ∈ {n}z = ⟨p, q⟩)
7		vex 2863	. . . . . . . . . . . . 13 ⊢ n ∈ V
8		opeq2 4580	. . . . . . . . . . . . . 14 ⊢ (q = n → ⟨p, q⟩ = ⟨p, n⟩)
9	8	eqeq2d 2364	. . . . . . . . . . . . 13 ⊢ (q = n → (z = ⟨p, q⟩ ↔ z = ⟨p, n⟩))
10	7, 9	rexsn 3769	. . . . . . . . . . . 12 ⊢ (∃q ∈ {n}z = ⟨p, q⟩ ↔ z = ⟨p, n⟩)
11	10	exbii 1582	. . . . . . . . . . 11 ⊢ (∃p∃q ∈ {n}z = ⟨p, q⟩ ↔ ∃p z = ⟨p, n⟩)
12	5, 6, 11	3bitri 262	. . . . . . . . . 10 ⊢ (z ∈ (V × {n}) ↔ ∃p z = ⟨p, n⟩)
13	12	anbi1i 676	. . . . . . . . 9 ⊢ ((z ∈ (V × {n}) ∧ z1st x) ↔ (∃p z = ⟨p, n⟩ ∧ z1st x))
14	4, 13	bitri 240	. . . . . . . 8 ⊢ ((z1st x ∧ z ∈ (V × {n})) ↔ (∃p z = ⟨p, n⟩ ∧ z1st x))
15		exancom 1586	. . . . . . . . 9 ⊢ (∃p(z1st x ∧ z = ⟨p, n⟩) ↔ ∃p(z = ⟨p, n⟩ ∧ z1st x))
16		19.41v 1901	. . . . . . . . 9 ⊢ (∃p(z = ⟨p, n⟩ ∧ z1st x) ↔ (∃p z = ⟨p, n⟩ ∧ z1st x))
17	15, 16	bitri 240	. . . . . . . 8 ⊢ (∃p(z1st x ∧ z = ⟨p, n⟩) ↔ (∃p z = ⟨p, n⟩ ∧ z1st x))
18	14, 17	bitr4i 243	. . . . . . 7 ⊢ ((z1st x ∧ z ∈ (V × {n})) ↔ ∃p(z1st x ∧ z = ⟨p, n⟩))
19		vex 2863	. . . . . . . . . . . 12 ⊢ x ∈ V
20	19	br1st 4859	. . . . . . . . . . 11 ⊢ (z1st x ↔ ∃q z = ⟨x, q⟩)
21	20	anbi1i 676	. . . . . . . . . 10 ⊢ ((z1st x ∧ z = ⟨p, n⟩) ↔ (∃q z = ⟨x, q⟩ ∧ z = ⟨p, n⟩))
22		19.41v 1901	. . . . . . . . . 10 ⊢ (∃q(z = ⟨x, q⟩ ∧ z = ⟨p, n⟩) ↔ (∃q z = ⟨x, q⟩ ∧ z = ⟨p, n⟩))
23	21, 22	bitr4i 243	. . . . . . . . 9 ⊢ ((z1st x ∧ z = ⟨p, n⟩) ↔ ∃q(z = ⟨x, q⟩ ∧ z = ⟨p, n⟩))
24	23	exbii 1582	. . . . . . . 8 ⊢ (∃p(z1st x ∧ z = ⟨p, n⟩) ↔ ∃p∃q(z = ⟨x, q⟩ ∧ z = ⟨p, n⟩))
25		eqeq1 2359	. . . . . . . . . . . . 13 ⊢ (z = ⟨p, n⟩ → (z = ⟨x, q⟩ ↔ ⟨p, n⟩ = ⟨x, q⟩))
26		opth 4603	. . . . . . . . . . . . 13 ⊢ (⟨p, n⟩ = ⟨x, q⟩ ↔ (p = x ∧ n = q))
27	25, 26	syl6bb 252	. . . . . . . . . . . 12 ⊢ (z = ⟨p, n⟩ → (z = ⟨x, q⟩ ↔ (p = x ∧ n = q)))
28	27	pm5.32ri 619	. . . . . . . . . . 11 ⊢ ((z = ⟨x, q⟩ ∧ z = ⟨p, n⟩) ↔ ((p = x ∧ n = q) ∧ z = ⟨p, n⟩))
29		equcom 1680	. . . . . . . . . . . . 13 ⊢ (n = q ↔ q = n)
30	29	anbi2i 675	. . . . . . . . . . . 12 ⊢ ((p = x ∧ n = q) ↔ (p = x ∧ q = n))
31	30	anbi1i 676	. . . . . . . . . . 11 ⊢ (((p = x ∧ n = q) ∧ z = ⟨p, n⟩) ↔ ((p = x ∧ q = n) ∧ z = ⟨p, n⟩))
32		opeq2 4580	. . . . . . . . . . . . . . 15 ⊢ (n = q → ⟨p, n⟩ = ⟨p, q⟩)
33	32	equcoms 1681	. . . . . . . . . . . . . 14 ⊢ (q = n → ⟨p, n⟩ = ⟨p, q⟩)
34	33	adantl 452	. . . . . . . . . . . . 13 ⊢ ((p = x ∧ q = n) → ⟨p, n⟩ = ⟨p, q⟩)
35	34	eqeq2d 2364	. . . . . . . . . . . 12 ⊢ ((p = x ∧ q = n) → (z = ⟨p, n⟩ ↔ z = ⟨p, q⟩))
36	35	pm5.32i 618	. . . . . . . . . . 11 ⊢ (((p = x ∧ q = n) ∧ z = ⟨p, n⟩) ↔ ((p = x ∧ q = n) ∧ z = ⟨p, q⟩))
37	28, 31, 36	3bitri 262	. . . . . . . . . 10 ⊢ ((z = ⟨x, q⟩ ∧ z = ⟨p, n⟩) ↔ ((p = x ∧ q = n) ∧ z = ⟨p, q⟩))
38		df-3an 936	. . . . . . . . . 10 ⊢ ((p = x ∧ q = n ∧ z = ⟨p, q⟩) ↔ ((p = x ∧ q = n) ∧ z = ⟨p, q⟩))
39	37, 38	bitr4i 243	. . . . . . . . 9 ⊢ ((z = ⟨x, q⟩ ∧ z = ⟨p, n⟩) ↔ (p = x ∧ q = n ∧ z = ⟨p, q⟩))
40	39	2exbii 1583	. . . . . . . 8 ⊢ (∃p∃q(z = ⟨x, q⟩ ∧ z = ⟨p, n⟩) ↔ ∃p∃q(p = x ∧ q = n ∧ z = ⟨p, q⟩))
41	24, 40	bitri 240	. . . . . . 7 ⊢ (∃p(z1st x ∧ z = ⟨p, n⟩) ↔ ∃p∃q(p = x ∧ q = n ∧ z = ⟨p, q⟩))
42		opeq1 4579	. . . . . . . . 9 ⊢ (p = x → ⟨p, q⟩ = ⟨x, q⟩)
43	42	eqeq2d 2364	. . . . . . . 8 ⊢ (p = x → (z = ⟨p, q⟩ ↔ z = ⟨x, q⟩))
44		opeq2 4580	. . . . . . . . 9 ⊢ (q = n → ⟨x, q⟩ = ⟨x, n⟩)
45	44	eqeq2d 2364	. . . . . . . 8 ⊢ (q = n → (z = ⟨x, q⟩ ↔ z = ⟨x, n⟩))
46	19, 7, 43, 45	ceqsex2v 2897	. . . . . . 7 ⊢ (∃p∃q(p = x ∧ q = n ∧ z = ⟨p, q⟩) ↔ z = ⟨x, n⟩)
47	18, 41, 46	3bitri 262	. . . . . 6 ⊢ ((z1st x ∧ z ∈ (V × {n})) ↔ z = ⟨x, n⟩)
48	2, 3, 47	3bitri 262	. . . . 5 ⊢ (x◡(1st ↾ (V × {n}))z ↔ z = ⟨x, n⟩)
49	48	anbi1i 676	. . . 4 ⊢ ((x◡(1st ↾ (V × {n}))z ∧ z AddC y) ↔ (z = ⟨x, n⟩ ∧ z AddC y))
50	49	exbii 1582	. . 3 ⊢ (∃z(x◡(1st ↾ (V × {n}))z ∧ z AddC y) ↔ ∃z(z = ⟨x, n⟩ ∧ z AddC y))
51	19, 7	opex 4589	. . . 4 ⊢ ⟨x, n⟩ ∈ V
52		breq1 4643	. . . 4 ⊢ (z = ⟨x, n⟩ → (z AddC y ↔ ⟨x, n⟩ AddC y))
53	51, 52	ceqsexv 2895	. . 3 ⊢ (∃z(z = ⟨x, n⟩ ∧ z AddC y) ↔ ⟨x, n⟩ AddC y)
54	1, 50, 53	3bitri 262	. 2 ⊢ (x( AddC ∘ ◡(1st ↾ (V × {n})))y ↔ ⟨x, n⟩ AddC y)
55	19, 7	braddcfn 5827	. 2 ⊢ (⟨x, n⟩ AddC y ↔ (x +c n) = y)
56		eqcom 2355	. 2 ⊢ ((x +c n) = y ↔ y = (x +c n))
57	54, 55, 56	3bitri 262	1 ⊢ (x( AddC ∘ ◡(1st ↾ (V × {n})))y ↔ y = (x +c n))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  {csn 3738   +_c cplc 4376  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722   × cxp 4771  ^◡ccnv 4772   ↾ cres 4775   AddC caddcfn 5746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-addcfn 5747

This theorem is referenced by:  addccan2nclem2  6265