Theorem mulsproplem4 28182

Description: Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of 𝐴 and a member of the old set of 𝐵 is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)

Hypotheses

Ref	Expression
mulsproplem.1	⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
mulsproplem4.1	⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))
mulsproplem4.2	⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))

Assertion

Ref	Expression
mulsproplem4	⊢ (𝜑 → (𝑋 ·s 𝑌) ∈ No )

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐵,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐷,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐸,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐹,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑋,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑌,𝑏,𝑐,𝑑,𝑒,𝑓

Allowed substitution hints:   𝜑(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝑌(𝑎)

Proof of Theorem mulsproplem4

Step	Hyp	Ref	Expression
1		mulsproplem.1	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
2		mulsproplem4.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))
3	2	oldnod 27910	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
4		mulsproplem4.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))
5	4	oldnod 27910	. . 3 ⊢ (𝜑 → 𝑌 ∈ No )
6		0no 27872	. . . 4 ⊢ 0s ∈ No
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No )
8		bday0 27874	. . . . . . . . . . . 12 ⊢ ( bday ‘ 0s ) = ∅
9	8, 8	oveq12i 7397	. . . . . . . . . . 11 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
10		0elon 6390	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
11		naddrid 8642	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
12	10, 11	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
13	9, 12	eqtri 2779	. . . . . . . . . 10 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
14	13, 13	uneq12i 4114	. . . . . . . . 9 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (∅ ∪ ∅)
15		un0 4342	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
16	14, 15	eqtri 2779	. . . . . . . 8 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = ∅
17	16, 16	uneq12i 4114	. . . . . . 7 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = (∅ ∪ ∅)
18	17, 15	eqtri 2779	. . . . . 6 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
19	18	uneq2i 4113	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ ∅)
20		un0 4342	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ ∅) = (( bday ‘𝑋) +no ( bday ‘𝑌))
21	19, 20	eqtri 2779	. . . 4 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = (( bday ‘𝑋) +no ( bday ‘𝑌))
22		oldbdayim 27952	. . . . . . 7 ⊢ (𝑋 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
23	2, 22	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
24		oldbdayim 27952	. . . . . . 7 ⊢ (𝑌 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
25	4, 24	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
26		bdayon 27815	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
27		bdayon 27815	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
28		naddel12 8659	. . . . . . 7 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑋) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑌) ∈ ( bday ‘𝐵)) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
29	26, 27, 28	mp2an 700	. . . . . 6 ⊢ ((( bday ‘𝑋) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑌) ∈ ( bday ‘𝐵)) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
30	23, 25, 29	syl2anc 592	. . . . 5 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
31		elun1 4129	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
32	30, 31	syl 17	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
33	21, 32	eqeltrid 2860	. . 3 ⊢ (𝜑 → ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
34	1, 3, 5, 7, 7, 7, 7, 33	mulsproplem1 28179	. 2 ⊢ (𝜑 → ((𝑋 ·s 𝑌) ∈ No ∧ (( 0s <s 0s ∧ 0s <s 0s ) → (( 0s ·s 0s ) -s ( 0s ·s 0s )) <s (( 0s ·s 0s ) -s ( 0s ·s 0s )))))
35	34	simpld 497	1 ⊢ (𝜑 → (𝑋 ·s 𝑌) ∈ No )

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∀wral 3070   ∪ cun 3897  ∅c0 4280   class class class wbr 5094  Oncon0 6335  ‘cfv 6510  (class class class)co 7385   +no cnadd 8623   No csur 27674   <s clts 27675   bday cbday 27676   0_s c0s 27868   O cold 27886   -_s csubs 28083   ·_s cmuls 28169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-1o 8425  df-2o 8426  df-nadd 8624  df-no 27677  df-lts 27678  df-bday 27679  df-slts 27821  df-cuts 27823  df-0s 27870  df-made 27890  df-old 27891

This theorem is referenced by:  mulsproplem5  28183  mulsproplem6  28184  mulsproplem7  28185  mulsproplem8  28186  mulsproplem9  28187