Theorem mulsproplem3 28513

Description: Lemma for surreal multiplication. Under the inductive hypothesis, the product of 𝐴 itself 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))))))
mulsproplem3.1	⊢ (𝜑 → 𝐴 ∈ No)
mulsproplem3.2	⊢ (𝜑 → 𝑌 ∈ (O‘(bday‘𝐵)))

Assertion

Ref	Expression
mulsproplem3	⊢ (𝜑 → (𝐴 ·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   𝑌,𝑏,𝑐,𝑑,𝑒,𝑓

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

Proof of Theorem mulsproplem3

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		mulsproplem3.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No)
3		mulsproplem3.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (O‘(bday‘𝐵)))
4	3	oldnod 28242	. . 3 ⊢ (𝜑 → 𝑌 ∈ No)
5		0no 28204	. . . 4 ⊢ 0s ∈ No
6	5	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No)
7		bday0 28206	. . . . . . . . . . . 12 ⊢ (bday‘ 0s ) = ∅
8	7, 7	oveq12i 7436	. . . . . . . . . . 11 ⊢ ((bday‘ 0s ) +no (bday‘ 0s )) = (∅ +no ∅)
9		0elon 6421	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
10		naddrid 8700	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
11	9, 10	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
12	8, 11	eqtri 2783	. . . . . . . . . 10 ⊢ ((bday‘ 0s ) +no (bday‘ 0s )) = ∅
13	12, 12	uneq12i 4164	. . . . . . . . 9 ⊢ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) = (∅ ∪ ∅)
14		un0 4343	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
15	13, 14	eqtri 2783	. . . . . . . 8 ⊢ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) = ∅
16	15, 15	uneq12i 4164	. . . . . . 7 ⊢ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s )))) = (∅ ∪ ∅)
17	16, 14	eqtri 2783	. . . . . 6 ⊢ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s )))) = ∅
18	17	uneq2i 4163	. . . . 5 ⊢ (((bday‘𝐴) +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‘𝑌)) ∪ ∅)
19		un0 4343	. . . . 5 ⊢ (((bday‘𝐴) +no (bday‘𝑌)) ∪ ∅) = ((bday‘𝐴) +no (bday‘𝑌))
20	18, 19	eqtri 2783	. . . 4 ⊢ (((bday‘𝐴) +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‘𝑌))
21		oldbdayim 28284	. . . . . . 7 ⊢ (𝑌 ∈ (O‘(bday‘𝐵)) → (bday‘𝑌) ∈ (bday‘𝐵))
22	3, 21	syl 18	. . . . . 6 ⊢ (𝜑 → (bday‘𝑌) ∈ (bday‘𝐵))
23		bdayon 28147	. . . . . . 7 ⊢ (bday‘𝑌) ∈ On
24		bdayon 28147	. . . . . . 7 ⊢ (bday‘𝐵) ∈ On
25		bdayon 28147	. . . . . . 7 ⊢ (bday‘𝐴) ∈ On
26		naddel2 8705	. . . . . . 7 ⊢ (((bday‘𝑌) ∈ On ∧ (bday‘𝐵) ∈ On ∧ (bday‘𝐴) ∈ On) → ((bday‘𝑌) ∈ (bday‘𝐵) ↔ ((bday‘𝐴) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵))))
27	23, 24, 25, 26	mp3an 1490	. . . . . 6 ⊢ ((bday‘𝑌) ∈ (bday‘𝐵) ↔ ((bday‘𝐴) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
28	22, 27	sylib 221	. . . . 5 ⊢ (𝜑 → ((bday‘𝐴) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
29		elun1 4179	. . . . 5 ⊢ (((bday‘𝐴) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵)) → ((bday‘𝐴) +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))))))
30	28, 29	syl 18	. . . 4 ⊢ (𝜑 → ((bday‘𝐴) +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))))))
31	20, 30	eqeltrid 2864	. . 3 ⊢ (𝜑 → (((bday‘𝐴) +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))))))
32	1, 2, 4, 6, 6, 6, 6, 31	mulsproplem1 28511	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝑌) ∈ No ∧ (( 0s <s 0s ∧ 0s <s 0s ) → (( 0s ·s 0s ) −s ( 0s ·s 0s )) <s (( 0s ·s 0s ) −s ( 0s ·s 0s )))))
33	32	simpld 500	1 ⊢ (𝜑 → (𝐴 ·s 𝑌) ∈ No)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∪ cun 4148  ∅c0 4277   class class class wbr 5102  Oncon0 6365  ‘cfv 6543  (class class class)co 7424   +no cnadd 8681  Nocsur 28006   <s clts 28007  bdaycbday 28008   0_s c0s 28200  Ocold 28218   −_s csubs 28415   ·_s cmuls 28501

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

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 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-1o 8483  df-2o 8484  df-nadd 8682  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-0s 28202  df-made 28222  df-old 28223

This theorem is used by:  mulsproplem5  28515  mulsproplem6  28516  mulsproplem7  28517  mulsproplem8  28518  mulsproplem9  28519  mulsproplem14  28524