Theorem xpcdaen 9005

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
xpcdaen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))

Proof of Theorem xpcdaen

Step	Hyp	Ref	Expression
1		enrefg 7987	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
2	1	3ad2ant1 1082	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ≈ 𝐴)
3		simp2 1062	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑊)
4		0ex 4790	. . . . . . 7 ⊢ ∅ ∈ V
5		xpsneng 8045	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
6	3, 4, 5	sylancl 694	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × {∅}) ≈ 𝐵)
7	6	ensymd 8007	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ≈ (𝐵 × {∅}))
8		xpen 8123	. . . . 5 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
9	2, 7, 8	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
10		simp3 1063	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
11		1on 7567	. . . . . . 7 ⊢ 1𝑜 ∈ On
12		xpsneng 8045	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
13	10, 11, 12	sylancl 694	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
14	13	ensymd 8007	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ≈ (𝐶 × {1𝑜}))
15		xpen 8123	. . . . 5 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐶 ≈ (𝐶 × {1𝑜})) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
16	2, 14, 15	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
17		xp01disj 7576	. . . . . . 7 ⊢ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
18	17	xpeq2i 5136	. . . . . 6 ⊢ (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = (𝐴 × ∅)
19		xpindi 5255	. . . . . 6 ⊢ (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜})))
20		xp0 5552	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
21	18, 19, 20	3eqtr3i 2652	. . . . 5 ⊢ ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅
22	21	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅)
23		cdaenun 8996	. . . 4 ⊢ (((𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})) ∧ (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})) ∧ ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
24	9, 16, 22, 23	syl3anc 1326	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
25		cdaval 8992	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
26	25	3adant1 1079	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
27	26	xpeq2d 5139	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))))
28		xpundi 5171	. . . 4 ⊢ (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜})))
29	27, 28	syl6eq 2672	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
30	24, 29	breqtrrd 4681	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵 +𝑐 𝐶)))
31	30	ensymd 8007	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  Oncon0 5723  (class class class)co 6650  1_𝑜c1o 7553   ≈ cen 7952   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-cda 8990

This theorem is referenced by: (None)