Theorem divmulsw 28185

Description: Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. Later, when we prove precsex 28210, we can eliminate the existence hypothesis (see divmuls 28213). (Contributed by Scott Fenton, 12-Mar-2025.)

Assertion

Ref	Expression
divmulsw	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem divmulsw

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divsval 28181	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → (𝐴 /su 𝐶) = (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴))
2	1	eqeq1d 2738	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
3	2	3expb 1121	. . . 4 ⊢ ((𝐴 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
4	3	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
5	4	adantr 480	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
6		simpl2 1194	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → 𝐵 ∈ No )
7		simp3l 1203	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ∈ No )
8		simp3r 1204	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ≠ 0s )
9		simp1 1137	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐴 ∈ No )
10	7, 8, 9	3jca 1129	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → (𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No ))
11		noreceuw 28183	. . . 4 ⊢ (((𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No ) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
12	10, 11	sylan 581	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
13		oveq2 7375	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐶 ·s 𝑦) = (𝐶 ·s 𝐵))
14	13	eqeq1d 2738	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐶 ·s 𝑦) = 𝐴 ↔ (𝐶 ·s 𝐵) = 𝐴))
15	14	riota2 7349	. . 3 ⊢ ((𝐵 ∈ No ∧ ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
16	6, 12, 15	syl2anc 585	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
17	5, 16	bitr4d 282	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  ∃!wreu 3340  ℩crio 7323  (class class class)co 7367   No csur 27603   0_s c0s 27797   1_s c1s 27798   ·_s cmuls 28098   /_su cdivs 28179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-muls 28099  df-divs 28180

This theorem is referenced by:  divmulswd  28186  divs1  28196  divmuls  28213