Theorem caov4d 5627

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovd.1	⊢ (φ → A ∈ 𝑆)
caovd.2	⊢ (φ → B ∈ 𝑆)
caovd.3	⊢ (φ → 𝐶 ∈ 𝑆)
caovd.com	⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) = (y𝐹x))
caovd.ass	⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))
caovd.4	⊢ (φ → 𝐷 ∈ 𝑆)
caovd.cl	⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆)

Assertion

Ref	Expression
caov4d	⊢ (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((A𝐹𝐶)𝐹(B𝐹𝐷)))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   x,𝐷,y,z   φ,x,y,z   x,𝐹,y,z   x,𝑆,y,z

Proof of Theorem caov4d

Step	Hyp	Ref	Expression
1		caovd.2	. . . 4 ⊢ (φ → B ∈ 𝑆)
2		caovd.3	. . . 4 ⊢ (φ → 𝐶 ∈ 𝑆)
3		caovd.4	. . . 4 ⊢ (φ → 𝐷 ∈ 𝑆)
4		caovd.com	. . . 4 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) = (y𝐹x))
5		caovd.ass	. . . 4 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))
6	1, 2, 3, 4, 5	caov12d 5624	. . 3 ⊢ (φ → (B𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(B𝐹𝐷)))
7	6	oveq2d 5471	. 2 ⊢ (φ → (A𝐹(B𝐹(𝐶𝐹𝐷))) = (A𝐹(𝐶𝐹(B𝐹𝐷))))
8		caovd.1	. . 3 ⊢ (φ → A ∈ 𝑆)
9		caovd.cl	. . . 4 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆)
10	9, 2, 3	caovcld 5596	. . 3 ⊢ (φ → (𝐶𝐹𝐷) ∈ 𝑆)
11	5, 8, 1, 10	caovassd 5602	. 2 ⊢ (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = (A𝐹(B𝐹(𝐶𝐹𝐷))))
12	9, 1, 3	caovcld 5596	. . 3 ⊢ (φ → (B𝐹𝐷) ∈ 𝑆)
13	5, 8, 2, 12	caovassd 5602	. 2 ⊢ (φ → ((A𝐹𝐶)𝐹(B𝐹𝐷)) = (A𝐹(𝐶𝐹(B𝐹𝐷))))
14	7, 11, 13	3eqtr4d 2079	1 ⊢ (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((A𝐹𝐶)𝐹(B𝐹𝐷)))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  (class class class)co 5455

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458

This theorem is referenced by:  caov411d  5628  caov42d  5629  ecopovtrn  6139  ecopovtrng  6142  addcmpblnq  6351  mulcmpblnq  6352  ordpipqqs  6358  distrnqg  6371  ltsonq  6382  ltanqg  6384  ltmnqg  6385  addcmpblnq0  6426  mulcmpblnq0  6427  distrnq0  6442  prarloclemlo  6477  addlocprlemeqgt  6515  addcanprleml  6588  recexprlem1ssl  6605  recexprlem1ssu  6606  mulcmpblnrlemg  6668  distrsrg  6687  ltasrg  6698  mulgt0sr  6704  axdistr  6758