Theorem decmul10add 9592

Description: A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)

Hypotheses

Ref	Expression
decmul10add.1	|- A e. NN0
decmul10add.2	|- B e. NN0
decmul10add.3	|- M e. NN0
decmul10add.4	|- E = ( M x. A )
decmul10add.5	|- F = ( M x. B )

Assertion

Ref	Expression
decmul10add	|- ( M x. ; A B ) = (; E 0 + F )

Proof of Theorem decmul10add

Step	Hyp	Ref	Expression
1		dfdec10 9527	. . 3 |- ; A B = ( (; 1 0 x. A ) + B )
2	1	oveq2i 5968	. 2 |- ( M x. ; A B ) = ( M x. ( (; 1 0 x. A ) + B ) )
3		decmul10add.3	. . . 4 |- M e. NN0
4	3	nn0cni 9327	. . 3 |- M e. CC
5		10nn0 9541	. . . . 5 |- ; 1 0 e. NN0
6		decmul10add.1	. . . . 5 |- A e. NN0
7	5, 6	nn0mulcli 9353	. . . 4 |- (; 1 0 x. A ) e. NN0
8	7	nn0cni 9327	. . 3 |- (; 1 0 x. A ) e. CC
9		decmul10add.2	. . . 4 |- B e. NN0
10	9	nn0cni 9327	. . 3 |- B e. CC
11	4, 8, 10	adddii 8102	. 2 |- ( M x. ( (; 1 0 x. A ) + B ) ) = ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) )
12	5	nn0cni 9327	. . . . 5 |- ; 1 0 e. CC
13	6	nn0cni 9327	. . . . 5 |- A e. CC
14	4, 12, 13	mul12i 8238	. . . 4 |- ( M x. (; 1 0 x. A ) ) = (; 1 0 x. ( M x. A ) )
15	3, 6	nn0mulcli 9353	. . . . 5 |- ( M x. A ) e. NN0
16	15	dec0u 9544	. . . 4 |- (; 1 0 x. ( M x. A ) ) = ; ( M x. A ) 0
17		decmul10add.4	. . . . . 6 |- E = ( M x. A )
18	17	eqcomi 2210	. . . . 5 |- ( M x. A ) = E
19	18	deceq1i 9530	. . . 4 |- ; ( M x. A ) 0 = ; E 0
20	14, 16, 19	3eqtri 2231	. . 3 |- ( M x. (; 1 0 x. A ) ) = ; E 0
21		decmul10add.5	. . . 4 |- F = ( M x. B )
22	21	eqcomi 2210	. . 3 |- ( M x. B ) = F
23	20, 22	oveq12i 5969	. 2 |- ( ( M x. (; 1 0 x. A ) ) + ( M x. B ) ) = (; E 0 + F )
24	2, 11, 23	3eqtri 2231	1 |- ( M x. ; A B ) = (; E 0 + F )

Syntax hints:   = wceq 1373   e. wcel 2177  (class class class)co 5957   0cc0 7945   1c1 7946   + caddc 7948   x. cmul 7950   NN0cn0 9315  ;cdc 9524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-sub 8265  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-9 9122  df-n0 9316  df-dec 9525

This theorem is referenced by: (None)