Theorem xpdjuen 10173

Description: Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
xpdjuen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 ⊔ 𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))

Proof of Theorem xpdjuen

Step	Hyp	Ref	Expression
1		enrefg 8979	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
2	1	3ad2ant1 1130	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ≈ 𝐴)
3		0ex 5300	. . . . . . 7 ⊢ ∅ ∈ V
4		simp2 1134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑊)
5		xpsnen2g 9064	. . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵)
6	3, 4, 5	sylancr 586	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ≈ 𝐵)
7	6	ensymd 9000	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ≈ ({∅} × 𝐵))
8		xpen 9139	. . . . 5 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐵 ≈ ({∅} × 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
9	2, 7, 8	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
10		1on 8476	. . . . . . 7 ⊢ 1o ∈ On
11		simp3 1135	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
12		xpsnen2g 9064	. . . . . . 7 ⊢ ((1o ∈ On ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
13	10, 11, 12	sylancr 586	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
14	13	ensymd 9000	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ≈ ({1o} × 𝐶))
15		xpen 9139	. . . . 5 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐶 ≈ ({1o} × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
16	2, 14, 15	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
17		xp01disjl 8490	. . . . . . 7 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
18	17	xpeq2i 5696	. . . . . 6 ⊢ (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (𝐴 × ∅)
19		xpindi 5826	. . . . . 6 ⊢ (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶)))
20		xp0 6150	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
21	18, 19, 20	3eqtr3i 2762	. . . . 5 ⊢ ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅
22	21	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅)
23		djuenun 10164	. . . 4 ⊢ (((𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)) ∧ (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)) ∧ ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶))))
24	9, 16, 22, 23	syl3anc 1368	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶))))
25		df-dju 9895	. . . . 5 ⊢ (𝐵 ⊔ 𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
26	25	xpeq2i 5696	. . . 4 ⊢ (𝐴 × (𝐵 ⊔ 𝐶)) = (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
27		xpundi 5737	. . . 4 ⊢ (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
28	26, 27	eqtri 2754	. . 3 ⊢ (𝐴 × (𝐵 ⊔ 𝐶)) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
29	24, 28	breqtrrdi 5183	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵 ⊔ 𝐶)))
30	29	ensymd 9000	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 ⊔ 𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∪ cun 3941   ∩ cin 3942  ∅c0 4317  {csn 4623   class class class wbr 5141   × cxp 5667  Oncon0 6357  1_oc1o 8457   ≈ cen 8935   ⊔ cdju 9892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1st 7971  df-2nd 7972  df-1o 8464  df-er 8702  df-en 8939  df-dom 8940  df-dju 9895

This theorem is referenced by: (None)