Theorem caofinvl 7659

Description: Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofinv.3	⊢ (𝜑 → 𝐵 ∈ 𝑊)
caofinv.4	⊢ (𝜑 → 𝑁:𝑆⟶𝑆)
caofinv.5	⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))))
caofinvl.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵)

Assertion

Ref	Expression
caofinvl	⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝐴 × {𝐵}))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑣,𝐴   𝑣,𝐹,𝑥   𝑥,𝑁,𝑣   𝑣,𝑆   𝜑,𝑣

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑣)   𝑅(𝑣)   𝐺(𝑣)   𝑉(𝑥,𝑣)   𝑊(𝑥,𝑣)

Proof of Theorem caofinvl

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		caofinv.5	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))))
3		caofinv.4	. . . . . . . 8 ⊢ (𝜑 → 𝑁:𝑆⟶𝑆)
4	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑁:𝑆⟶𝑆)
5		caofref.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
6	5	ffvelcdmda 7032	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝑆)
7	4, 6	ffvelcdmd 7033	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑁‘(𝐹‘𝑣)) ∈ 𝑆)
8	2, 7	fmpt3d 7064	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
9	8	ffvelcdmda 7032	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
10	5	ffvelcdmda 7032	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
11		fvex 6847	. . . . . . 7 ⊢ (𝑁‘(𝐹‘𝑣)) ∈ V
12		eqid 2740	. . . . . . 7 ⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))
13	11, 12	fnmpti 6635	. . . . . 6 ⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴
14	2	fneq1d 6585	. . . . . 6 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴))
15	13, 14	mpbiri 259	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
16		dffn5 6892	. . . . 5 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
17	15, 16	sylib 219	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
18	5	feqmptd 6902	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
19	1, 9, 10, 17, 18	offval2 7647	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))))
20	2	fveq1d 6836	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑤) = ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤))
21		2fveq3 6839	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑁‘(𝐹‘𝑣)) = (𝑁‘(𝐹‘𝑤)))
22		fvex 6847	. . . . . . . 8 ⊢ (𝑁‘(𝐹‘𝑤)) ∈ V
23	21, 12, 22	fvmpt 6942	. . . . . . 7 ⊢ (𝑤 ∈ 𝐴 → ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤) = (𝑁‘(𝐹‘𝑤)))
24	20, 23	sylan9eq 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝑁‘(𝐹‘𝑤)))
25	24	oveq1d 7378	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)))
26		fveq2 6834	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑁‘𝑥) = (𝑁‘(𝐹‘𝑤)))
27		id 22	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤))
28	26, 27	oveq12d 7381	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑁‘𝑥)𝑅𝑥) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)))
29	28	eqeq1d 2742	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑁‘𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵))
30		caofinvl.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
31	30	ralrimiva 3132	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
32	31	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵)
33	29, 32, 10	rspcdva 3568	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵)
34	25, 33	eqtrd 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = 𝐵)
35	34	mpteq2dva 5172	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) = (𝑤 ∈ 𝐴 ↦ 𝐵))
36	19, 35	eqtrd 2775	. 2 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ 𝐵))
37		fconstmpt 5687	. 2 ⊢ (𝐴 × {𝐵}) = (𝑤 ∈ 𝐴 ↦ 𝐵)
38	36, 37	eqtr4di 2793	1 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝐴 × {𝐵}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {csn 4562   ↦ cmpt 5160   × cxp 5623   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ∘_f cof 7625

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 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627

This theorem is referenced by:  grpvlinv  22388  lflnegl  39575