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Mirrors > Home > NFE Home > Th. List > mucnc | GIF version |
Description: Cardinal multiplication in terms of cardinality. Theorem XI.2.27 of [Rosser] p. 378. (Contributed by SF, 10-Mar-2015.) |
Ref | Expression |
---|---|
mucnc.1 | ⊢ A ∈ V |
mucnc.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
mucnc | ⊢ ( Nc A ·c Nc B) = Nc (A × B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mucnc.1 | . . . 4 ⊢ A ∈ V | |
2 | 1 | ncelncsi 6121 | . . 3 ⊢ Nc A ∈ NC |
3 | mucnc.2 | . . . 4 ⊢ B ∈ V | |
4 | 3 | ncelncsi 6121 | . . 3 ⊢ Nc B ∈ NC |
5 | ovmuc 6130 | . . 3 ⊢ (( Nc A ∈ NC ∧ Nc B ∈ NC ) → ( Nc A ·c Nc B) = {x ∣ ∃y ∈ Nc A∃z ∈ Nc Bx ≈ (y × z)}) | |
6 | 2, 4, 5 | mp2an 653 | . 2 ⊢ ( Nc A ·c Nc B) = {x ∣ ∃y ∈ Nc A∃z ∈ Nc Bx ≈ (y × z)} |
7 | df-nc 6101 | . . 3 ⊢ Nc (A × B) = [(A × B)] ≈ | |
8 | dfec2 5948 | . . 3 ⊢ [(A × B)] ≈ = {x ∣ (A × B) ≈ x} | |
9 | elnc 6125 | . . . . . . . 8 ⊢ (y ∈ Nc A ↔ y ≈ A) | |
10 | elnc 6125 | . . . . . . . 8 ⊢ (z ∈ Nc B ↔ z ≈ B) | |
11 | 9, 10 | anbi12i 678 | . . . . . . 7 ⊢ ((y ∈ Nc A ∧ z ∈ Nc B) ↔ (y ≈ A ∧ z ≈ B)) |
12 | ensym 6037 | . . . . . . 7 ⊢ (x ≈ (y × z) ↔ (y × z) ≈ x) | |
13 | 11, 12 | anbi12i 678 | . . . . . 6 ⊢ (((y ∈ Nc A ∧ z ∈ Nc B) ∧ x ≈ (y × z)) ↔ ((y ≈ A ∧ z ≈ B) ∧ (y × z) ≈ x)) |
14 | 13 | 2exbii 1583 | . . . . 5 ⊢ (∃y∃z((y ∈ Nc A ∧ z ∈ Nc B) ∧ x ≈ (y × z)) ↔ ∃y∃z((y ≈ A ∧ z ≈ B) ∧ (y × z) ≈ x)) |
15 | r2ex 2652 | . . . . 5 ⊢ (∃y ∈ Nc A∃z ∈ Nc Bx ≈ (y × z) ↔ ∃y∃z((y ∈ Nc A ∧ z ∈ Nc B) ∧ x ≈ (y × z))) | |
16 | 1 | enrflx 6035 | . . . . . . 7 ⊢ A ≈ A |
17 | 3 | enrflx 6035 | . . . . . . 7 ⊢ B ≈ B |
18 | breq1 4642 | . . . . . . . . . 10 ⊢ (y = A → (y ≈ A ↔ A ≈ A)) | |
19 | breq1 4642 | . . . . . . . . . 10 ⊢ (z = B → (z ≈ B ↔ B ≈ B)) | |
20 | 18, 19 | bi2anan9 843 | . . . . . . . . 9 ⊢ ((y = A ∧ z = B) → ((y ≈ A ∧ z ≈ B) ↔ (A ≈ A ∧ B ≈ B))) |
21 | xpeq12 4803 | . . . . . . . . . 10 ⊢ ((y = A ∧ z = B) → (y × z) = (A × B)) | |
22 | 21 | breq1d 4649 | . . . . . . . . 9 ⊢ ((y = A ∧ z = B) → ((y × z) ≈ x ↔ (A × B) ≈ x)) |
23 | 20, 22 | anbi12d 691 | . . . . . . . 8 ⊢ ((y = A ∧ z = B) → (((y ≈ A ∧ z ≈ B) ∧ (y × z) ≈ x) ↔ ((A ≈ A ∧ B ≈ B) ∧ (A × B) ≈ x))) |
24 | 1, 3, 23 | spc2ev 2947 | . . . . . . 7 ⊢ (((A ≈ A ∧ B ≈ B) ∧ (A × B) ≈ x) → ∃y∃z((y ≈ A ∧ z ≈ B) ∧ (y × z) ≈ x)) |
25 | 16, 17, 24 | mpanl12 663 | . . . . . 6 ⊢ ((A × B) ≈ x → ∃y∃z((y ≈ A ∧ z ≈ B) ∧ (y × z) ≈ x)) |
26 | xpen 6055 | . . . . . . . . 9 ⊢ ((y ≈ A ∧ z ≈ B) → (y × z) ≈ (A × B)) | |
27 | ensym 6037 | . . . . . . . . 9 ⊢ ((y × z) ≈ (A × B) ↔ (A × B) ≈ (y × z)) | |
28 | 26, 27 | sylib 188 | . . . . . . . 8 ⊢ ((y ≈ A ∧ z ≈ B) → (A × B) ≈ (y × z)) |
29 | entr 6038 | . . . . . . . 8 ⊢ (((A × B) ≈ (y × z) ∧ (y × z) ≈ x) → (A × B) ≈ x) | |
30 | 28, 29 | sylan 457 | . . . . . . 7 ⊢ (((y ≈ A ∧ z ≈ B) ∧ (y × z) ≈ x) → (A × B) ≈ x) |
31 | 30 | exlimivv 1635 | . . . . . 6 ⊢ (∃y∃z((y ≈ A ∧ z ≈ B) ∧ (y × z) ≈ x) → (A × B) ≈ x) |
32 | 25, 31 | impbii 180 | . . . . 5 ⊢ ((A × B) ≈ x ↔ ∃y∃z((y ≈ A ∧ z ≈ B) ∧ (y × z) ≈ x)) |
33 | 14, 15, 32 | 3bitr4ri 269 | . . . 4 ⊢ ((A × B) ≈ x ↔ ∃y ∈ Nc A∃z ∈ Nc Bx ≈ (y × z)) |
34 | 33 | abbii 2465 | . . 3 ⊢ {x ∣ (A × B) ≈ x} = {x ∣ ∃y ∈ Nc A∃z ∈ Nc Bx ≈ (y × z)} |
35 | 7, 8, 34 | 3eqtrri 2378 | . 2 ⊢ {x ∣ ∃y ∈ Nc A∃z ∈ Nc Bx ≈ (y × z)} = Nc (A × B) |
36 | 6, 35 | eqtri 2373 | 1 ⊢ ( Nc A ·c Nc B) = Nc (A × B) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 ∃wrex 2615 Vcvv 2859 class class class wbr 4639 × cxp 4770 (class class class)co 5525 [cec 5945 ≈ 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: muccl 6132 muccom 6134 mucass 6135 muc0 6142 mucid1 6143 addcdi 6250 muc0or 6252 |
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