Theorem mulsproplem4 28517

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 28245	. . 3 ⊢ (𝜑 → 𝑋 ∈ No)
4		mulsproplem4.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (O‘(bday‘𝐵)))
5	4	oldnod 28245	. . 3 ⊢ (𝜑 → 𝑌 ∈ No)
6		0no 28207	. . . 4 ⊢ 0s ∈ No
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No)
8		bday0 28209	. . . . . . . . . . . 12 ⊢ (bday‘ 0s ) = ∅
9	8, 8	oveq12i 7439	. . . . . . . . . . 11 ⊢ ((bday‘ 0s ) +no (bday‘ 0s )) = (∅ +no ∅)
10		0elon 6424	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
11		naddrid 8704	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
12	10, 11	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
13	9, 12	eqtri 2786	. . . . . . . . . 10 ⊢ ((bday‘ 0s ) +no (bday‘ 0s )) = ∅
14	13, 13	uneq12i 4167	. . . . . . . . 9 ⊢ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) = (∅ ∪ ∅)
15		un0 4346	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
16	14, 15	eqtri 2786	. . . . . . . 8 ⊢ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) = ∅
17	16, 16	uneq12i 4167	. . . . . . 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 2786	. . . . . 6 ⊢ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s )))) = ∅
19	18	uneq2i 4166	. . . . 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 4346	. . . . 5 ⊢ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ∅) = ((bday‘𝑋) +no (bday‘𝑌))
21	19, 20	eqtri 2786	. . . 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 28287	. . . . . . 7 ⊢ (𝑋 ∈ (O‘(bday‘𝐴)) → (bday‘𝑋) ∈ (bday‘𝐴))
23	2, 22	syl 18	. . . . . 6 ⊢ (𝜑 → (bday‘𝑋) ∈ (bday‘𝐴))
24		oldbdayim 28287	. . . . . . 7 ⊢ (𝑌 ∈ (O‘(bday‘𝐵)) → (bday‘𝑌) ∈ (bday‘𝐵))
25	4, 24	syl 18	. . . . . 6 ⊢ (𝜑 → (bday‘𝑌) ∈ (bday‘𝐵))
26		bdayon 28150	. . . . . . 7 ⊢ (bday‘𝐴) ∈ On
27		bdayon 28150	. . . . . . 7 ⊢ (bday‘𝐵) ∈ On
28		naddel12 8721	. . . . . . 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 705	. . . . . 6 ⊢ (((bday‘𝑋) ∈ (bday‘𝐴) ∧ (bday‘𝑌) ∈ (bday‘𝐵)) → ((bday‘𝑋) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
30	23, 25, 29	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((bday‘𝑋) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
31		elun1 4182	. . . . 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 18	. . . 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 2867	. . 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 28514	. 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 500	1 ⊢ (𝜑 → (𝑋 ·s 𝑌) ∈ No)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3079   ∪ cun 4151  ∅c0 4280   class class class wbr 5105  Oncon0 6368  ‘cfv 6546  (class class class)co 7427   +no cnadd 8685  Nocsur 28009   <s clts 28010  bdaycbday 28011   0_s c0s 28203  Ocold 28221   −_s csubs 28418   ·_s cmuls 28504

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-se 5608  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-1o 8487  df-2o 8488  df-nadd 8686  df-no 28012  df-lts 28013  df-bday 28014  df-slts 28156  df-cuts 28158  df-0s 28205  df-made 28225  df-old 28226

This theorem is used by:  mulsproplem5  28518  mulsproplem6  28519  mulsproplem7  28520  mulsproplem8  28521  mulsproplem9  28522