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Mirrors > Home > NFE Home > Th. List > caovdig | GIF version |
Description: Convert an operation distributive law to class notation. (Contributed by set.mm contributors, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.) |
Ref | Expression |
---|---|
caovdig.1 | ⊢ ((φ ∧ (x ∈ S ∧ y ∈ S ∧ z ∈ S)) → (xG(yFz)) = ((xGy)F(xGz))) |
Ref | Expression |
---|---|
caovdig | ⊢ ((φ ∧ (A ∈ S ∧ B ∈ S ∧ C ∈ S)) → (AG(BFC)) = ((AGB)F(AGC))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovdig.1 | . . 3 ⊢ ((φ ∧ (x ∈ S ∧ y ∈ S ∧ z ∈ S)) → (xG(yFz)) = ((xGy)F(xGz))) | |
2 | 1 | ralrimivvva 2707 | . 2 ⊢ (φ → ∀x ∈ S ∀y ∈ S ∀z ∈ S (xG(yFz)) = ((xGy)F(xGz))) |
3 | oveq1 5530 | . . . 4 ⊢ (x = A → (xG(yFz)) = (AG(yFz))) | |
4 | oveq1 5530 | . . . . 5 ⊢ (x = A → (xGy) = (AGy)) | |
5 | oveq1 5530 | . . . . 5 ⊢ (x = A → (xGz) = (AGz)) | |
6 | 4, 5 | oveq12d 5540 | . . . 4 ⊢ (x = A → ((xGy)F(xGz)) = ((AGy)F(AGz))) |
7 | 3, 6 | eqeq12d 2367 | . . 3 ⊢ (x = A → ((xG(yFz)) = ((xGy)F(xGz)) ↔ (AG(yFz)) = ((AGy)F(AGz)))) |
8 | oveq1 5530 | . . . . 5 ⊢ (y = B → (yFz) = (BFz)) | |
9 | 8 | oveq2d 5538 | . . . 4 ⊢ (y = B → (AG(yFz)) = (AG(BFz))) |
10 | oveq2 5531 | . . . . 5 ⊢ (y = B → (AGy) = (AGB)) | |
11 | 10 | oveq1d 5537 | . . . 4 ⊢ (y = B → ((AGy)F(AGz)) = ((AGB)F(AGz))) |
12 | 9, 11 | eqeq12d 2367 | . . 3 ⊢ (y = B → ((AG(yFz)) = ((AGy)F(AGz)) ↔ (AG(BFz)) = ((AGB)F(AGz)))) |
13 | oveq2 5531 | . . . . 5 ⊢ (z = C → (BFz) = (BFC)) | |
14 | 13 | oveq2d 5538 | . . . 4 ⊢ (z = C → (AG(BFz)) = (AG(BFC))) |
15 | oveq2 5531 | . . . . 5 ⊢ (z = C → (AGz) = (AGC)) | |
16 | 15 | oveq2d 5538 | . . . 4 ⊢ (z = C → ((AGB)F(AGz)) = ((AGB)F(AGC))) |
17 | 14, 16 | eqeq12d 2367 | . . 3 ⊢ (z = C → ((AG(BFz)) = ((AGB)F(AGz)) ↔ (AG(BFC)) = ((AGB)F(AGC)))) |
18 | 7, 12, 17 | rspc3v 2964 | . 2 ⊢ ((A ∈ S ∧ B ∈ S ∧ C ∈ S) → (∀x ∈ S ∀y ∈ S ∀z ∈ S (xG(yFz)) = ((xGy)F(xGz)) → (AG(BFC)) = ((AGB)F(AGC)))) |
19 | 2, 18 | mpan9 455 | 1 ⊢ ((φ ∧ (A ∈ S ∧ B ∈ S ∧ C ∈ S)) → (AG(BFC)) = ((AGB)F(AGC))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∧ w3a 934 = wceq 1642 ∈ wcel 1710 ∀wral 2614 (class class class)co 5525 |
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-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-br 4640 df-fv 4795 df-ov 5526 |
This theorem is referenced by: caovdi 5634 |
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