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Mirrors > Home > NFE Home > Th. List > muccom | GIF version |
Description: Cardinal multiplication commutes. Theorem XI.2.28 of [Rosser] p. 378. (Contributed by SF, 10-Mar-2015.) |
Ref | Expression |
---|---|
muccom | ⊢ ((A ∈ NC ∧ B ∈ NC ) → (A ·c B) = (B ·c A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elncs 6119 | . . . 4 ⊢ (A ∈ NC ↔ ∃x A = Nc x) | |
2 | elncs 6119 | . . . 4 ⊢ (B ∈ NC ↔ ∃y B = Nc y) | |
3 | 1, 2 | anbi12i 678 | . . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) ↔ (∃x A = Nc x ∧ ∃y B = Nc y)) |
4 | eeanv 1913 | . . 3 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) ↔ (∃x A = Nc x ∧ ∃y B = Nc y)) | |
5 | 3, 4 | bitr4i 243 | . 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) ↔ ∃x∃y(A = Nc x ∧ B = Nc y)) |
6 | vex 2862 | . . . . . . 7 ⊢ x ∈ V | |
7 | vex 2862 | . . . . . . 7 ⊢ y ∈ V | |
8 | 6, 7 | xpcomen 6052 | . . . . . 6 ⊢ (x × y) ≈ (y × x) |
9 | 6, 7 | xpex 5115 | . . . . . . 7 ⊢ (x × y) ∈ V |
10 | 9 | eqnc 6127 | . . . . . 6 ⊢ ( Nc (x × y) = Nc (y × x) ↔ (x × y) ≈ (y × x)) |
11 | 8, 10 | mpbir 200 | . . . . 5 ⊢ Nc (x × y) = Nc (y × x) |
12 | 6, 7 | mucnc 6131 | . . . . 5 ⊢ ( Nc x ·c Nc y) = Nc (x × y) |
13 | 7, 6 | mucnc 6131 | . . . . 5 ⊢ ( Nc y ·c Nc x) = Nc (y × x) |
14 | 11, 12, 13 | 3eqtr4i 2383 | . . . 4 ⊢ ( Nc x ·c Nc y) = ( Nc y ·c Nc x) |
15 | oveq12 5532 | . . . 4 ⊢ ((A = Nc x ∧ B = Nc y) → (A ·c B) = ( Nc x ·c Nc y)) | |
16 | oveq12 5532 | . . . . 5 ⊢ ((B = Nc y ∧ A = Nc x) → (B ·c A) = ( Nc y ·c Nc x)) | |
17 | 16 | ancoms 439 | . . . 4 ⊢ ((A = Nc x ∧ B = Nc y) → (B ·c A) = ( Nc y ·c Nc x)) |
18 | 14, 15, 17 | 3eqtr4a 2411 | . . 3 ⊢ ((A = Nc x ∧ B = Nc y) → (A ·c B) = (B ·c A)) |
19 | 18 | exlimivv 1635 | . 2 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) → (A ·c B) = (B ·c A)) |
20 | 5, 19 | sylbi 187 | 1 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (A ·c B) = (B ·c A)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 class class class wbr 4639 × cxp 4770 (class class class)co 5525 ≈ cen 6028 NC cncs 6088 Nc cnc 6091 ·c cmuc 6092 |
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 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-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 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-csb 3137 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-iun 3971 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-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-pprod 5738 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-cross 5764 df-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-en 6029 df-ncs 6098 df-nc 6101 df-muc 6102 |
This theorem is referenced by: addcdir 6251 muc0or 6252 lemuc2 6254 |
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